
theorem
  for S,T being non empty RelStr, f be set holds (f is Connection of S,T
iff f is Connection of S~,T) & (f is Connection of S,T iff f is Connection of S
  ,T~) & (f is Connection of S,T iff f is Connection of S~,T~)
proof
  let S,T be non empty RelStr;
A1: now
    let S,S1,T,T1 be RelStr;
    assume
A2: the carrier of S = the carrier of S1 & the carrier of T = the
    carrier of T1;
    let a be Connection of S,T;
    consider f being Function of S,T, g being Function of T,S such that
A3: a = [f,g] by WAYBEL_1:def 9;
    reconsider g as Function of T1,S1 by A2;
    reconsider f as Function of S1,T1 by A2;
    a = [f,g] by A3;
    hence a is Connection of S1,T1;
  end;
  S~ = RelStr(#the carrier of S, (the InternalRel of S)~#) & T~ = RelStr(#
    the carrier of T, (the InternalRel of T)~#);
  hence thesis by A1;
end;
