
theorem Th2:
  for R1,R2 being real RelStr st the carrier of R1 = the carrier of
  R2 holds the RelStr of R1 = the RelStr of R2
proof
  let R1, R2 be real RelStr such that
A1: the carrier of R1 = the carrier of R2;
  set X = the carrier of R1;
  the InternalRel of R1 = the InternalRel of R2
  proof
    let a,b be object;
A2: X c= REAL by Def1;
    hereby
      assume
A3:   [a,b] in the InternalRel of R1;
      then
A4:   a in X & b in X by ZFMISC_1:87;
      then reconsider a9 = a, b9 = b as Element of REAL by A2;
      a9 <= b9 by A3,A4,Def1;
      hence [a,b] in the InternalRel of R2 by A1,A4,Def1;
    end;
    assume
A5: [a,b] in the InternalRel of R2;
    then
A6: a in X & b in X by A1,ZFMISC_1:87;
    then reconsider a9 = a, b9 = b as Element of REAL by A2;
    a9 <= b9 by A1,A5,A6,Def1;
    hence thesis by A6,Def1;
  end;
  hence thesis by A1;
end;
