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 Th39:
  F is_isomorphism_of R,S implies F" is_isomorphism_of S,R
proof
  assume
A1: F is_isomorphism_of R,S;
  then
A2: F is one-to-one;
A3: rng F = field S by A1;
  hence
A4: dom(F") = field S by A2,FUNCT_1:33;
 dom F = field R by A1;
  hence rng(F") = field R by A2,FUNCT_1:33;
  thus F" is one-to-one by A2;
  let a,b;
  thus [a,b] in S implies a in field S & b in field S & [F".a,F".b] in R
  proof
A5: dom F = rng(F") by A2,FUNCT_1:33;
    assume
A6: [a,b] in S;
    hence
A7: a in field S & b in field S by RELAT_1:15;
    then
A8: F".a in rng(F") & F".b in rng(F") by A4,FUNCT_1:def 3;
    a = F.(F".a) & b = F.(F".b) by A3,A2,A7,FUNCT_1:35;
    hence thesis by A1,A6,A5,A8;
  end;
  assume that
A9: a in field S & b in field S and
A10: [F".a,F".b] in R;
  F.(F".a) = a & F.(F".b) = b by A3,A2,A9,FUNCT_1:35;
  hence thesis by A1,A10;
end;
