
theorem Th5:
  for R being non empty RelStr,
      a, b being Element of R st
    a in UAp ({b}) holds [a,b] in the InternalRel of R
  proof
    let R be non empty RelStr;
    let a, b be Element of R;
    assume a in UAp ({b}); then
    consider x being Element of R such that
A1: x = a & Class (the InternalRel of R,x) meets {b};
    consider y being object such that
A2: y in Class (the InternalRel of R,x) /\ {b} by A1,XBOOLE_0:def 1;
    y in Class (the InternalRel of R,x) & y in {b} by XBOOLE_0:def 4,A2; then
    y = b & y in Class (the InternalRel of R,x) by TARSKI:def 1;
    hence thesis by A1,RELAT_1:169;
  end;
