theorem Th91:
  x in f.:A iff ex y st y in A & f.y=x
proof
  thus x in f.:A implies ex y st y in A & f.y=x
  proof
    assume x in f.:A;
    then ex y being object st y in dom f & y in A & x=f.y by FUNCT_1:def 6;
    hence thesis;
  end;
  given y such that
A1: y in A & f.y=x;
  dom f = the carrier of AFS by FUNCT_2:52;
  hence thesis by A1,FUNCT_1:def 6;
end;
