reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem Th36:
  for A,B,C,D being Real st A>0 & C>0 holds AffineMap(A,B,C,D) is onto
proof
  let A,B,C,D be Real such that
A1: A>0 & C>0;
  thus rng AffineMap(A,B,C,D) c= the carrier of TOP-REAL 2;
  let e be object;
  assume e in the carrier of TOP-REAL 2;
  then reconsider q1 = e as Point of TOP-REAL 2;
  set q2 = AffineMap(1/A,-B/A,1/C,-D/C).q1;
A2: the carrier of TOP-REAL 2 = REAL 2 by EUCLID:22;
  AffineMap(A,B,C,D).q2 = (AffineMap(A,B,C,D)*AffineMap(1/A,-B/A,1/C,-D/C)
  ).q1 by FUNCT_2:15
    .= (id REAL 2).q1 by A1,Th34
    .= q1 by A2;
  hence thesis by FUNCT_2:4;
end;
