
theorem Th36:
  for R being connected non empty Poset,a,b being Element of FinPoset R
  holds [a,b] in the InternalRel of FinPoset R iff
  ex x,y being Element of Fin the carrier of R st a = x & b = y & (x = {} or
  x<>{} & y<>{} & PosetMax x <> PosetMax y &
  [PosetMax x,PosetMax y] in the InternalRel of R or
  x<>{} & y<>{} & PosetMax x = PosetMax y &
  [x\{PosetMax x},y\{PosetMax y}] in FinOrd R)
proof
  let R be connected non empty Poset, a,b be Element of FinPoset R;
  set CR = the carrier of R;
  reconsider x=a, y=b as Element of Fin CR;
  hereby
    assume
A1: [a,b] in the InternalRel of FinPoset R;
    take x,y;
    thus a = x & b = y;
    thus x = {} or x<>{} & y<>{} & PosetMax x <> PosetMax y &
    [PosetMax x,PosetMax y] in the InternalRel of R or
    x<>{} & y<>{} & PosetMax x = PosetMax y &
    [x\{PosetMax x},y\{PosetMax y}] in FinOrd R by A1,Th32;
  end;
  assume ex x,y being Element of Fin the carrier of R st a = x & b = y &
  ( x = {} or x<>{} & y<>{} & PosetMax x <> PosetMax y &
  [PosetMax x,PosetMax y] in the InternalRel of R or
  x<>{} & y<>{} & PosetMax x = PosetMax y &
  [x\{PosetMax x},y\{PosetMax y}] in FinOrd R );
  hence thesis by Th32;
end;
