reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve f for RingMorphismStr;
reserve G,H,G1,G2,G3,G4 for Ring;
reserve F for RingMorphism;
reserve V for Ring_DOMAIN;

theorem Th15:
  for x,y1,y2 being object st GO x,y1 & GO x,y2 holds y1 = y2
proof
  let x,y1,y2 be object such that
A1: GO x,y1 and
A2: GO x,y2;
  consider a1,a2,a3,a4,a5,a6 being set such that
A3: x = [[a1,a2,a3,a4],a5,a6] and
A4: ex G being strict Ring st y1 = G & a1 = the carrier of G & a2 = the
addF of G & a3 = comp G & a4 = 0.G & a5 = the multF of G & a6 = 1.G by A1;
  consider b1,b2,b3,b4,b5,b6 being set such that
A5: x = [[b1,b2,b3,b4],b5,b6] and
A6: ex G being strict Ring st y2 = G & b1 = the carrier of G & b2 = the
addF of G & b3 = comp G & b4 = 0.G & b5 = the multF of G & b6 = 1.G by A2;
  consider G2 being strict Ring such that
A7: y2 = G2 and
A8: b1 = the carrier of G2 & b2 = the addF of G2 and
  b3 = comp G2 and
A9: b4 = 0.G2 and
A10: b5 = the multF of G2 & b6 = 1.G2 by A6;
  consider G1 being strict Ring such that
A11: y1 = G1 and
A12: a1 = the carrier of G1 & a2 = the addF of G1 and
  a3 = comp G1 and
A13: a4 = 0.G1 and
A14: a5 = the multF of G1 & a6 = 1.G1 by A4;
A15: the multF of G1 = the multF of G2 & 1.G1 = 1.G2 by A3,A5,A14,A10,
XTUPLE_0:3;
A16: [a1,a2,a3,a4] = [b1,b2,b3,b4] by A3,A5,XTUPLE_0:3;
  then
  the carrier of G1 = the carrier of G2 & the addF of G1 = the addF of G2
  by A12,A8,XTUPLE_0:5;
  hence thesis by A11,A13,A7,A9,A16,A15,XTUPLE_0:5;
end;
