reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem Th27:
  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 being set such that
A3: x = [a1,a2,a3,a4] and
A4: ex G being strict AddGroup st y1 = G & a1 = the carrier of G & a2 =
  the addF of G & a3 = comp G & a4 = 0.G by A1;
  consider G1 being strict AddGroup such that
A5: y1 = G1 and
A6: a1 = the carrier of G1 & a2 = the addF of G1 and
  a3 = comp G1 and
A7: a4 = 0.G1 by A4;
  consider b1,b2,b3,b4 being set such that
A8: x = [b1,b2,b3,b4] and
A9: ex G being strict AddGroup st y2 = G & b1 = the carrier of G & b2 =
  the addF of G & b3 = comp G & b4 = 0.G by A2;
  consider G2 being strict AddGroup such that
A10: y2 = G2 and
A11: b1 = the carrier of G2 & b2 = the addF of G2 and
  b3 = comp G2 and
A12: b4 = 0.G2 by A9;
  the carrier of G1 = the carrier of G2 & the addF of G1 = the addF of G2
  by A3,A8,A6,A11,XTUPLE_0:5;
  hence thesis by A3,A8,A5,A7,A10,A12,XTUPLE_0:5;
end;
