reserve x,y for set,
  n,m for Nat,
  r,s for Real;
reserve f, g for Function;

theorem Th30: :: RFINSEQ:17
  for f,g be FinSequence,x be set st x in dom g &
  f,g are_fiberwise_equipotent ex y be set st y in dom g & f.x=g.y
proof
  let f,g be FinSequence,x be set;
  assume that
A1: x in dom g and
A2: f,g are_fiberwise_equipotent;
  consider P be Permutation of dom g such that
A3: f = g*P by A2,Th4;
  take y=P.x;
  thus y in dom g by A1,FUNCT_2:5;
  thus thesis by A1,A3,FUNCT_2:15;
end;
