reserve

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

theorem Th18:
  for S1,S2 being IncProjStr for F being IncProjMap over S1,S2 for
K being Subset of the Points of S1 holds F.:K = {B where B is POINT of S2:ex A
  being POINT of S1 st (A 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 S1;
  set Image = {B where B is POINT of S2:ex A being POINT of S1 st (A in K & F.
  A = B)};
A1: F.:K c= Image
  proof
    let b be object;
    assume b in F.:K;
    then consider a being object such that
A2: a in dom (the point-map of F) and
A3: a in K and
A4: b = (the point-map of F).a by FUNCT_1:def 6;
    consider A being POINT of S1 such that
A5: a = A by A2;
    b in the Points of S2 by A2,A4,FUNCT_2:5;
    then consider B1 being POINT of S2 such that
A6: b = B1;
    F.A = B1 by A4,A5,A6;
    hence thesis by A3,A4,A5;
  end;
  Image c= F.:K
  proof
    let b be object;
    assume b in Image;
    then
A7: ex B being POINT of S2 st B = b & ex A being POINT of S1 st A in K & F
    .A = B;
    the Points of S1 = dom (the point-map of F) by FUNCT_2:def 1;
    hence thesis by A7,FUNCT_1:def 6;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
