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 Th43:
  F is_isomorphism_of R,S implies ( R is reflexive implies S is
  reflexive ) & ( R is transitive implies S is transitive ) & ( R is connected
implies S is connected ) & ( R is antisymmetric implies S is antisymmetric ) &
  ( R is well_founded implies S is well_founded )
proof
  assume
A1: F is_isomorphism_of R,S;
  then
A2: dom F = field R;
A3: rng F = field S by A1;
A4: F is one-to-one by A1;
  then
A5: rng(F") = dom F & dom(F") = rng F by FUNCT_1:33;
  thus R is reflexive implies S is reflexive
  proof
    assume
A6: R is reflexive;
    now
      let a;
      assume
A7:   a in field S;
      then F".a in field R by A2,A3,A5,FUNCT_1:def 3;
      then
A8:   [F".a,F".a] in R by A6,Lm1;
      a = F.(F".a) by A3,A4,A7,FUNCT_1:35;
      hence [a,a] in S by A1,A8;
    end;
    hence thesis by Lm1;
  end;
  thus R is transitive implies S is transitive
  proof
    assume
A9: R is transitive;
    now
      let a,b,c;
      assume that
A10:  [a,b] in S and
A11:  [b,c] in S;
A12:  c in field S by A11,RELAT_1:15;
      then
A13:  c = F.(F".c) by A3,A4,FUNCT_1:35;
      b in field S by A10,RELAT_1:15;
      then
A14:  b = F.(F".b) & F".b in field R by A2,A3,A4,A5,FUNCT_1:35,def 3;
      F".c in field R by A2,A3,A5,A12,FUNCT_1:def 3;
      then
A15:  [F".b,F".c] in R by A1,A11,A13,A14;
A16:  a in field S by A10,RELAT_1:15;
      then
A17:  a = F.(F".a) by A3,A4,FUNCT_1:35;
      F".a in field R by A2,A3,A5,A16,FUNCT_1:def 3;
      then [F".a,F".b] in R by A1,A10,A17,A14;
      then [F".a,F".c] in R by A9,A15,Lm2;
      hence [a,c] in S by A1,A17,A13;
    end;
    hence thesis by Lm2;
  end;
  thus R is connected implies S is connected
  proof
    assume
A18: R is connected;
    now
      let a,b;
      assume that
A19:  a in field S & b in field S and
A20:  a <> b;
A21:  a = F.(F".a) & b = F.(F".b) by A3,A4,A19,FUNCT_1:35;
      F".a in field R & F".b in field R by A2,A3,A5,A19,FUNCT_1:def 3;
      then [F".a,F".b] in R or [F".b,F".a] in R by A18,A20,A21,Lm4;
      hence [a,b] in S or [b,a] in S by A1,A21;
    end;
    hence thesis by Lm4;
  end;
  thus R is antisymmetric implies S is antisymmetric
  proof
    assume
A22: R is antisymmetric;
    now
      let a,b;
      assume that
A23:  [a,b] in S and
A24:  [b,a] in S;
A25:  a in field S by A23,RELAT_1:15;
      then
A26:  a = F.(F".a) by A3,A4,FUNCT_1:35;
A27:  b in field S by A23,RELAT_1:15;
      then
A28:  b = F.(F".b) by A3,A4,FUNCT_1:35;
A29:  F".b in field R by A2,A3,A5,A27,FUNCT_1:def 3;
      F".a in field R by A2,A3,A5,A25,FUNCT_1:def 3;
      then [F".a,F".b] in R & [F".b,F".a] in R by A1,A23,A24,A26,A28,A29;
      hence a = b by A22,A26,A28,Lm3;
    end;
    hence thesis by Lm3;
  end;
  assume
A30: for Y st Y c= field R & Y <> {} ex x st x in Y & R-Seg(x) misses Y;
  let Z;
  assume that
A31: Z c= field S and
A32: Z <> {};
A33: F"Z c= dom F by RELAT_1:132;
  then consider x such that
A34: x in F"Z and
A35: R-Seg(x) misses F"Z by A2,A3,A30,A31,A32,RELAT_1:139;
  take F.x;
  thus F.x in Z by A34,FUNCT_1:def 7;
  assume not thesis;
  then consider y being object such that
A36: y in S-Seg(F.x) and
A37: y in Z by XBOOLE_0:3;
A38: F".y in dom F by A3,A5,A31,A37,FUNCT_1:def 3;
A39: [y,F.x] in S by A36,Th1;
A40: y = F.(F".y) by A3,A4,A31,A37,FUNCT_1:35;
  then F".y in F"Z by A37,A38,FUNCT_1:def 7;
  then not F".y in R-Seg(x) by A35,XBOOLE_0:3;
  then not [F".y,x] in R or F".y = x by Th1;
  hence contradiction by A1,A33,A34,A36,A40,A38,A39,Th1;
end;
