
theorem Th81:
  for x1,y1, x2,y2 being set holds x1 U+ y1 \/ x2 U+ y2 = (x1 \/
  x2) U+ (y1 \/ y2)
proof
  let x1,y1, x2,y2 be set;
  set X1 = [:x1,{1}:], X2 = [:x2,{1}:];
  set Y1 = [:y1,{2}:], Y2 = [:y2,{2}:];
  set X = [:x1 \/ x2, {1}:], Y = [:y1 \/ y2, {2}:];
A1: X = X1 \/ X2 by ZFMISC_1:97;
A2: (x1 \/ x2) U+ (y1 \/ y2) = X \/ Y & Y = Y1 \/ Y2 by Th73,ZFMISC_1:97;
  x1 U+ y1 = X1 \/ Y1 & x2 U+ y2 = X2 \/ Y2 by Th73;
  hence x1 U+ y1 \/ x2 U+ y2 = X1 \/ Y1 \/ X2 \/ Y2 by XBOOLE_1:4
    .= X \/ Y1 \/ Y2 by A1,XBOOLE_1:4
    .= (x1 \/ x2) U+ (y1 \/ y2) by A2,XBOOLE_1:4;
end;
