
theorem Th73:
  for x,y being set holds x U+ y = [:x,{1}:] \/ [:y,{2}:]
proof
  let x,y be set;
  len <*x,y*> = 2 by FINSEQ_1:44;
  then
A1: dom <*x,y*> = {1,2} by FINSEQ_1:2,def 3;
  now
    let z be object;
A2: z`2 in {1,2} iff z`2 = 1 or z`2 = 2 by TARSKI:def 2;
A3: z in [:x,{1}:] iff z`1 in x & z`2 = 1 & z = [z`1,z`2] by MCART_1:13,21
,ZFMISC_1:106;
A4: z in [:y,{2}:] iff z`1 in y & z`2 = 2 & z = [z`1,z`2] by MCART_1:13,21
,ZFMISC_1:106;
    z in x U+ y iff z`2 in {1,2} & z`1 in <*x,y*>.(z`2) & z = [z`1,z`2] by A1,
CARD_3:22;
    hence z in x U+ y iff z in [:x,{1}:] \/ [:y,{2}:] by A2,A3,A4,XBOOLE_0:def
3;
  end;
  hence thesis by TARSKI:2;
end;
