reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th1:
  for a,b being set st a <> b & card a = n & card b = n holds card
  (a /\ b) in n & n + 1 c= card (a \/ b)
proof
  let a,b be set;
  assume that
A1: a <> b and
A2: card a = n and
A3: card b = n and
A4: not card (a /\ b) in n or not n + 1 c= card (a \/ b);
  n c= card (a /\ b) or card (a \/ b) in Segm(n + 1) by A4,ORDINAL1:16;
  then n c= card (a /\ b) or card (a \/ b) in succ Segm n by NAT_1:38;
  then
A5: n c= card (a /\ b) or card (a \/ b) c= n by ORDINAL1:22;
  n c= card(a \/ b) & card(a /\ b) c= n by A2,CARD_1:11,XBOOLE_1:7,17;
  then
A6: card a = card (a /\ b) & card (a /\ b) = card b or card (a \/ b) = card
  a & card (a \/ b) = card b by A2,A3,A5,XBOOLE_0:def 10;
A7: a c= a \/ b & b c= a \/ b by XBOOLE_1:7;
  a is finite set & b is finite set by A2,A3;
  then a = a /\ b & b = a /\ b or a = a \/ b & b = a \/ b by A7,A6,CARD_2:102
,XBOOLE_1:17;
  hence contradiction by A1;
end;
