reserve

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

theorem
  for S1,S2 being IncProjStr for F being IncProjMap over S1,S2 for K
  being Subset of the Points of S2 holds F"K = {A where A is POINT of S1:ex B
  being POINT of S2 st (B in K & F.A = B)}
proof
  let S1,S2 be IncProjStr;
  let F be IncProjMap over S1,S2;
  let K be Subset of the Points of S2;
  set Image = {A where A is POINT of S1:ex B being POINT of S2 st (B in K & F.
  A = B)};
A1: F"K c= Image
  proof
    let a be object;
    assume
A2: a in F"K;
    then consider A being POINT of S1 such that
A3: a = A;
A4: (the point-map of F).a in K by A2,FUNCT_1:def 7;
    then consider B1 being POINT of S2 such that
A5: (the point-map of F).a = B1;
    F.A = B1 by A3,A5;
    hence thesis by A4,A3;
  end;
  Image c= F"K
  proof
    let a be object;
    assume a in Image;
    then
A6: ex A being POINT of S1 st A = a & ex B being POINT of S2 st B in K & F.
    A = B;
    the Points of S1 = dom (the point-map of F) by FUNCT_2:def 1;
    hence thesis by A6,FUNCT_1:def 7;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
