
theorem Th12:
  for S, T being Subset of TOP-REAL 2 st S = {p where p is Point
of TOP-REAL 2 : p`2 >= 1 - 2 * p`1 } & T = {p where p is Point of TOP-REAL 2 :
  p`2 >= - p`1 } holds AffineMap (1, 0, 1/2, -1/2) .: S = T
proof
  set f = AffineMap (1,0,1/2,-1/2);
  set A = 1, B = 0, C = 1/2, D = -1/2;
  let S, T be Subset of TOP-REAL 2;
  assume that
A1: S = {p where p is Point of TOP-REAL 2 : p`2 >= 1 - 2 * p`1 } and
A2: T = {p where p is Point of TOP-REAL 2 : p`2 >= - p`1 };
  f .: S = T
  proof
    thus f .: S c= T
    proof
      let x be object;
      assume x in f .: S;
      then consider y being object such that
      y in dom f and
A3:   y in S and
A4:   x = f.y by FUNCT_1:def 6;
      consider p being Point of TOP-REAL 2 such that
A5:   y = p and
A6:   p`2 >= 1 - 2 * p`1 by A1,A3;
      set b = f.p;
      C * p`2 >= C * (1 - 2 * p`1) by A6,XREAL_1:64;
      then
A7:   C * p`2 + D >= C - p`1 + D by XREAL_1:6;
A8:   f.p = |[A*(p`1)+B,C*(p`2)+D]| by JGRAPH_2:def 2;
      then b`1 = A * (p`1) + B by EUCLID:52;
      then b`2 >= - b`1 by A8,A7,EUCLID:52;
      hence thesis by A2,A4,A5;
    end;
    let x be object;
    assume
A9: x in T;
    then
A10: ex p being Point of TOP-REAL 2 st x = p & p`2 >= - p`1 by A2;
    f is onto by JORDAN1K:36;
    then rng f = the carrier of TOP-REAL 2 by FUNCT_2:def 3;
    then consider y being object such that
A11: y in dom f and
A12: x = f.y by A9,FUNCT_1:def 3;
    reconsider y as Point of TOP-REAL 2 by A11;
    set b = f.y;
A13: f.y = |[A*(y`1)+B,C*(y`2)+D]| by JGRAPH_2:def 2;
    then b`1 = y`1 by EUCLID:52;
    then 2 * b`2 >= 2 * (- y`1) by A10,A12,XREAL_1:64;
    then
A14: 2 * b`2 + 1 >= 2 * (- y`1) + 1 by XREAL_1:6;
    b`2 = C * (y`2) + D by A13,EUCLID:52;
    then y`2 >= 1 - 2 * y`1 by A14;
    then y in S by A1;
    hence thesis by A11,A12,FUNCT_1:def 6;
  end;
  hence thesis;
end;
