reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve R for Ring;
reserve G,H for LeftMod of R;
reserve V for LeftMod_DOMAIN of R;

theorem Th4:
  for x,y1,y2 being object st GO x,y1,R & GO x,y2,R holds y1 = y2
proof
  let x,y1,y2 be object such that
A1: GO x,y1,R and
A2: GO x,y2,R;
  consider a1,a2 being object such that
A3: x = [a1,a2] and
A4: ex G being strict LeftMod of R st y1 = G & a1 = the addLoopStr of G
  & a2 = the lmult of G by A1;
  consider G1 being strict LeftMod of R such that
A5: y1 = G1 and
A6: a1 = the addLoopStr of G1 and
A7: a2 = the lmult of G1 by A4;
  consider b1,b2 being object such that
A8: x = [b1,b2] and
A9: ex G being strict LeftMod of R st y2 = G & b1 = the addLoopStr of G
  & b2 = the lmult of G by A2;
  consider G2 being strict LeftMod of R such that
A10: y2 = G2 and
A11: b1 = the addLoopStr of G2 and
A12: b2 = the lmult of G2 by A9;
  the addLoopStr of G1 = the addLoopStr of G2 by A3,A8,A6,A11,XTUPLE_0:1;
  hence thesis by A3,A8,A5,A7,A10,A12,XTUPLE_0:1;
end;
