
theorem
  for L being with_suprema Poset for S being join-inheriting non empty
full SubRelStr of L for x,y being Element of S, a,b be Element of L st a = x &
  b = y holds x"\/"y = a"\/"b
proof
  let L be with_suprema Poset;
  let S be join-inheriting non empty full SubRelStr of L;
  let x,y be Element of S, a,b be Element of L such that
A1: a = x & b = y;
A2: ex_sup_of {a,b},L by Th20;
  then "\/"({x,y},L) in the carrier of S by A1,Def17;
  then
A3: "\/"({x,y},S) = "\/"({x,y},L) by A1,A2,Th64;
  a"\/"b = sup {a,b } by Th41;
  hence thesis by A1,A3,Th41;
end;
