
theorem
  for S being non empty RelStr, T being non empty reflexive RelStr, x
  being Element of [:S,T:] holds proj1 uparrow x = uparrow x`1
proof
  let S be non empty RelStr, T be non empty reflexive RelStr, x be Element of
  [:S,T:];
A1: x`2 <= x`2;
  thus proj1 uparrow x c= uparrow x`1 by Th41;
  let a be object;
  assume
A2: a in uparrow x`1;
  then reconsider a9 = a as Element of S;
  a9 >= x`1 by A2,WAYBEL_0:18;
  then [a9,x`2] >= [x`1,x`2] by A1,YELLOW_3:11;
  then
A3: [a9,x`2] in uparrow [x`1,x`2] by WAYBEL_0:18;
  the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then x = [x`1,x`2] by MCART_1:21;
  hence thesis by A3,XTUPLE_0:def 12;
end;
