
theorem Th82:
  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_1:99;
A2: {1} misses {2} by ZFMISC_1:11;
  then Y1 misses X2 by ZFMISC_1:104;
  then
A3: Y = Y1 /\ Y2 & Y1 /\ X2 = {} by ZFMISC_1:99;
  X1 misses Y2 by A2,ZFMISC_1:104;
  then
A4: X1 /\ Y2 = {};
  x1 U+ y1 = X1 \/ Y1 & x2 U+ y2 = X2 \/ Y2 by Th73;
  hence (x1 U+ y1) /\ (x2 U+ y2) = ((X1 \/ Y1) /\ X2) \/ ((X1 \/ Y1) /\ Y2) by
XBOOLE_1:23
    .= (X \/ Y1 /\ X2) \/ ((X1 \/ Y1) /\ Y2) by A1,XBOOLE_1:23
    .= X \/ (X1 /\ Y2 \/ Y) by A3,XBOOLE_1:23
    .= (x1 /\ x2) U+ (y1 /\ y2) by A4,Th73;
end;
