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 Th48:
  R is well-ordering & Z c= field R & F is_isomorphism_of R,S
  implies F|Z is_isomorphism_of R |_2 Z,S |_2 (F.:Z) & R |_2 Z,S |_2 (F.:Z)
  are_isomorphic
proof
  assume that
A1: R is well-ordering and
A2: Z c= field R and
A3: F is_isomorphism_of R,S;
A4: F.:Z c= rng F by RELAT_1:111;
A5: F.:Z = field(S |_2 (F.:Z)) by A1,A3,A4,Th31,Th44;
A6: F is one-to-one by A3;
A7: Z = field(R |_2 Z) by A1,A2,Th31;
A8: dom F = field R by A3;
  thus F|Z is_isomorphism_of R |_2 Z,S |_2 (F.:Z)
  proof
    thus
A9: dom(F|Z) = field(R |_2 Z) by A2,A8,A7,RELAT_1:62;
    thus
A10: rng(F|Z) = field(S |_2 (F.:Z)) by A5,RELAT_1:115;
    thus F|Z is one-to-one by A6,FUNCT_1:52;
    let a,b;
    thus [a,b] in R |_2 Z implies a in field(R |_2 Z) & b in field(R |_2 Z) &
    [F|Z.a,F|Z.b] in S |_2 (F.:Z)
    proof
      assume
A11:  [a,b] in R |_2 Z;
      then [a,b] in R by XBOOLE_0:def 4;
      then
A12:  [F.a,F.b] in S by A3;
      thus
A13:  a in field(R |_2 Z) & b in field(R |_2 Z) by A11,RELAT_1:15;
      then F|Z.a in rng(F|Z) & F|Z.b in rng(F|Z) by A9,FUNCT_1:def 3;
      then
A14:  [F|Z.a,F|Z.b] in [:F.:Z,F.:Z:] by A5,A10,ZFMISC_1:87;
      F.a = F|Z.a & F.b = F|Z.b by A9,A13,FUNCT_1:47;
      hence thesis by A12,A14,XBOOLE_0:def 4;
    end;
    assume that
A15: a in field(R |_2 Z) & b in field(R |_2 Z) and
A16: [F|Z.a,F|Z.b] in S |_2 (F.:Z);
    F.a = F|Z.a & F.b = F|Z.b by A9,A15,FUNCT_1:47;
    then
A17: [F.a,F.b] in S by A16,XBOOLE_0:def 4;
A18: [a,b] in [:Z,Z:] by A7,A15,ZFMISC_1:87;
    a in field R & b in field R by A15,Th12;
    then [a,b] in R by A3,A17;
    hence thesis by A18,XBOOLE_0:def 4;
  end;
  hence thesis;
end;
