
theorem
  for S being non empty reflexive RelStr, T being non empty RelStr, x
  being Element of [:S,T:] holds proj2 downarrow x = downarrow x`2
proof
  let S be non empty reflexive RelStr, T be non empty RelStr, x be Element of
  [:S,T:];
A1: x`1 <= x`1;
  thus proj2 downarrow x c= downarrow x`2 by Th37;
  let b be object;
  assume
A2: b in downarrow x`2;
  then reconsider b9 = b as Element of T;
  b9 <= x`2 by A2,WAYBEL_0:17;
  then [x`1,b9] <= [x`1,x`2] by A1,YELLOW_3:11;
  then
A3: [x`1,b9] in downarrow [x`1,x`2] by WAYBEL_0:17;
  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 13;
end;
