
theorem
  for L be non empty transitive RelStr for S be join-closed non empty
Subset of L for x,y be Element of S st ex_sup_of {x,y},L holds ex_sup_of {x,y},
  subrelstr S & "\/"({x,y},subrelstr S) = "\/"({x,y},L)
proof
  let L be non empty transitive RelStr;
  let S be join-closed non empty Subset of L;
  let x,y be Element of S;
A1: x is Element of subrelstr S by YELLOW_0:def 15;
A2: y is Element of subrelstr S by YELLOW_0:def 15;
  assume
A3: ex_sup_of {x,y},L;
  subrelstr S is join-inheriting non empty full SubRelStr of L by Def2;
  then "\/"({x,y},L) in the carrier of subrelstr S by A1,A2,A3,YELLOW_0:def 17;
  hence thesis by A1,A2,A3,YELLOW_0:66;
end;
