
theorem Th32: :: CRS:
for n being Nat, x,y being set
 st x in n & y in n
  holds [x,y] in the InternalRel of CompleteRelStr n iff x <> y
proof
let n be Nat, x,y be set; assume
A1: x in n & y in n;
  hereby
    assume [x,y] in the InternalRel of CompleteRelStr n;
      then [x,y] in [:n,n:] \ id n by Def8;
      then not [x,y] in id n by XBOOLE_0:def 5;
    hence x <> y by A1,RELAT_1:def 10;
  end;
  assume x <> y;
  then [x,y] in [:n,n:] & not [x,y] in id n by A1,RELAT_1:def 10,ZFMISC_1:87;
  then [x,y] in [:n,n:] \ id n by XBOOLE_0:def 5;
  hence [x,y] in the InternalRel of CompleteRelStr n by Def8;
end;
