
theorem Th7:
for X being non empty finite set, P being a_partition of X
 st card P < card X
  ex p being set, x, y being object st p in P & x in p & y in p & x <> y
proof
  let X be non empty finite set, P be a_partition of X such that
A1: card P < card X;
A2: card P in Segm card X by A1,NAT_1:44;
  consider x,y being object such that
A3: x in X and
A4: y in X and
A5: x <> y and
A6: (proj P).x = (proj P).y by A2,FINSEQ_4:65;
  take p = (proj P).x;
  take x, y;
  thus p in P by A3,FUNCT_2:5;
  thus x in p & y in p by A3,A4,A6,EQREL_1:def 9;
  thus x <> y by A5;
end;
