reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th49:
  F is_isomorphism_of R,S implies for a st a in field R ex b st b
  in field S & F.:(R-Seg(a)) = S-Seg(b)
proof
  assume
A1: F is_isomorphism_of R,S;
  then
A2: dom F = field R;
  let a;
  assume
A3: a in field R;
  take b = F.a;
A4: rng F = field S by A1;
  hence b in field S by A3,A2,FUNCT_1:def 3;
A5: F is one-to-one by A1;
A6: for c being object holds c in S-Seg(b) implies c in F.:(R-Seg(a))
  proof let c be object;
    assume
A7: c in S-Seg(b);
    then
A8: c <> b by Th1;
A9: [c,b] in S by A7,Th1;
    then
A10: c in field S by RELAT_1:15;
    then
A11: c = F.(F".c) by A4,A5,FUNCT_1:35;
    rng(F") = dom F & dom(F") = rng F by A5,FUNCT_1:33;
    then
A12: F".c in field R by A2,A4,A10,FUNCT_1:def 3;
    then [F".c,a] in R by A1,A3,A9,A11;
    then F".c in R-Seg(a) by A8,A11,Th1;
    hence thesis by A2,A11,A12,FUNCT_1:def 6;
  end;
  for c being object holds c in F.:(R-Seg(a)) implies c in S-Seg(b)
  proof let c be object;
    assume c in F.:(R-Seg(a));
    then consider d being object such that
A13: d in dom F and
A14: d in R-Seg(a) and
A15: c = F.d by FUNCT_1:def 6;
    [d,a] in R by A14,Th1;
    then
A16: [c,b] in S by A1,A15;
    d <> a by A14,Th1;
    then c <> b by A3,A2,A5,A13,A15;
    hence thesis by A16,Th1;
  end;
  hence thesis by A6,TARSKI:2;
end;
