
theorem Th17:
  for S, T being bounded with_suprema with_infima antisymmetric
  RelStr, x, y being Element of [:S,T:] holds x is_a_complement_of y iff x`1
  is_a_complement_of y`1 & x`2 is_a_complement_of y`2
proof
  let S, T be bounded with_suprema with_infima antisymmetric RelStr, x, y be
  Element of [:S,T:];
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  hereby
    assume
A2: x is_a_complement_of y;
A3: y`1 "/\" x`1 = (y "/\" x)`1 by Th13
      .= (Bottom [:S,T:])`1 by A2
      .= [Bottom S,Bottom T]`1 by Th4
      .= Bottom S;
    y`1 "\/" x`1 = (y "\/" x)`1 by Th14
      .= (Top [:S,T:])`1 by A2
      .= [Top S,Top T]`1 by Th3
      .= Top S;
    hence x`1 is_a_complement_of y`1 by A3;
A4: y`2 "/\" x`2 = (y "/\" x)`2 by Th13
      .= (Bottom [:S,T:])`2 by A2
      .= [Bottom S,Bottom T]`2 by Th4
      .= Bottom T;
    y`2 "\/" x`2 = (y "\/" x)`2 by Th14
      .= (Top [:S,T:])`2 by A2
      .= [Top S,Top T]`2 by Th3
      .= Top T;
    hence x`2 is_a_complement_of y`2 by A4;
  end;
  assume that
A5: y`1 "\/" x`1 = Top S and
A6: y`1 "/\" x`1 = Bottom S and
A7: y`2 "\/" x`2 = Top T and
A8: y`2 "/\" x`2 = Bottom T;
A9: (y "\/" x)`2 = y`2 "\/" x`2 by Th14
    .= [Top S,Top T]`2 by A7;
  (y "\/" x)`1 = y`1 "\/" x`1 by Th14
    .= [Top S,Top T]`1 by A5;
  hence y "\/" x = [Top S,Top T] by A1,A9,DOMAIN_1:2
    .= Top [:S,T:] by Th3;
A10: (y "/\" x)`2 = y`2 "/\" x`2 by Th13
    .= [Bottom S,Bottom T]`2 by A8;
  (y "/\" x)`1 = y`1 "/\" x`1 by Th13
    .= [Bottom S,Bottom T]`1 by A6;
  hence y "/\" x = [Bottom S,Bottom T] by A1,A10,DOMAIN_1:2
    .= Bottom [:S,T:] by Th4;
end;
