  Context: GRB110709B is the first source for which Swift BAT triggered twice,
with a time separation of ~10 min. The first emission (Ep. 1) goes from 40s
before the 1{\deg} trigger up to 60s after it. The second (Ep. 2) goes from 35s
before the 2{\deg} trigger to 100s after it.[...] Within the Induced
Gravitational Collapse (IGC) model, we assume the progenitor to be a close
binary system composed of a core of an evolved star and a Neutron Star (NS).
The evolved star explodes as a Supernova (SN) and ejects material that is
partially accreted by the NS. We identify this process with Ep. 1. The
accretion process brings the NS over its critical mass, thus gravitationally
collapsing to a BH. This process leads to the GRB emission, Ep. 2.[...] Aims:
We analyze the spectra and time variability of Ep. 1 and 2 and compute the
relevant parameters of the binary progenitor[...] in the IGC paradigm. Methods:
We perform a time-resolved spectral analysis of Ep. 1 with a blackbody (BB)
plus a power-law (PL) spectral model. We analyze Ep. 2 within the Fireshell
model, identifying the Proper-GRB (P-GRB) and simulating the light curve and
spectrum. We establish the redshift to be z=0.75 [...]. Results: We find for
Ep. 1 a BB temperature following a broken PL with time, with the PL slopes at
early and late times \alpha=0 and \beta=-4+/-2, respectively, and a break at
t=41.21s. The total energy of Ep. 1 and 2 are E_{iso}^1=1.42x10^{53}erg and
E_{iso}^2=2.43x10^{52}erg, respectively. We find at transparency a Lorentz
factor \Gamma~173, laboratory radius of 6.04x10^{13}cm, P-GRB observed
temperature kT_{P-GRB}=12.36keV, baryon load B=0.0057 and P-GRB energy
E_{P-GRB}=3.44x10^{50}erg. [...] Conclusions: We interpret GRB110709B as a
member of the IGC sources, together with GRB970828, GRB090618 and GRB101023.
The XRT data during Ep. 2 offers an unprecedented tool for improving the
diagnostic of GRBs emission.
