
theorem
  for R1,R2,R3 being non empty RelStr for f1 being Function of R1,R3 st
  f1 is isomorphic for f2 being Function of R2,R3 st f2=f1 & f2 is isomorphic
  holds the RelStr of R1 = the RelStr of R2
proof
  let R1,R2,R3 be non empty RelStr;
  let f1 be Function of R1,R3;
  assume
A1: f1 is isomorphic;
  let f2 be Function of R2,R3;
  assume that
A2: f2=f1 and
A3: f2 is isomorphic;
A4: the carrier of R1 = rng (f2 qua Function") by A1,A2,WAYBEL_0:67
    .= the carrier of R2 by A3,WAYBEL_0:67;
A5: the InternalRel of R2 c= the InternalRel of R1
  proof
    let x1,x2 be object;
    assume
A6: [x1,x2] in the InternalRel of R2;
    then reconsider x19=x1,x29=x2 as Element of R2 by ZFMISC_1:87;
    reconsider y1=x19,y2=x29 as Element of R1 by A4;
    x19 <= x29 by A6,ORDERS_2:def 5;
    then f2.x19 <= f2.x29 by A3,WAYBEL_0:66;
    then y1 <= y2 by A1,A2,WAYBEL_0:66;
    hence thesis by ORDERS_2:def 5;
  end;
  the InternalRel of R1 c= the InternalRel of R2
  proof
    let x1,x2 be object;
    assume
A7: [x1,x2] in the InternalRel of R1;
    then reconsider x19=x1,x29=x2 as Element of R1 by ZFMISC_1:87;
    reconsider y1=x19,y2=x29 as Element of R2 by A4;
    x19 <= x29 by A7,ORDERS_2:def 5;
    then f1.x19 <= f1.x29 by A1,WAYBEL_0:66;
    then y1 <= y2 by A2,A3,WAYBEL_0:66;
    hence thesis by ORDERS_2:def 5;
  end;
  hence thesis by A4,A5,XBOOLE_0:def 10;
end;
