 reserve A for non empty Subset of REAL;

theorem
  for a,b being Real st a > 0 holds
  |. AffineMap (a,b) .| = (- (AffineMap(a,b) | ].-infty,(-b)/a.[))
   +* (AffineMap(a,b) | [.(-b)/a,+infty.[)
proof
 let a,b be Real;
  assume A1: a > 0;
  INF: -infty < (-b)/a & (-b)/a < +infty
     by XXREAL_0:9,XXREAL_0:12,XREAL_0:def 1;
  D1a: dom ((- (AffineMap(a,b) | ].-infty,(-b)/a.[ ))
   +* (AffineMap(a,b) | [.(-b)/a,+infty.[ ) )
  = dom ((- (AffineMap(a,b) | ].-infty,(-b)/a.[ )))
     \/ dom((AffineMap(a,b) | [.(-b)/a,+infty.[ ) ) by FUNCT_4:def 1
.= ].-infty,(-b)/a.[ \/ dom( AffineMap(a,b) | [.(-b)/a,+infty.[ )
    by FUNCT_2:def 1
.= ].-infty,(-b)/a.[ \/ [.(-b)/a,+infty.[ by FUNCT_2:def 1
.= REAL by XXREAL_1:224,XXREAL_1:173,INF;
  for x being object st x in dom (|. AffineMap (a,b) .|) holds
 |. AffineMap (a,b) .| .x
 =( (- (AffineMap(a,b) | ].-infty,(-b)/a.[ ))
   +* (AffineMap(a,b) | [.(-b)/a,+infty.[ )) . x
  proof
   let x be object;
   assume x in dom (|. AffineMap (a,b) .|); then
   reconsider x as Real;
   A5: |. AffineMap (a,b) .| .x = |. AffineMap (a,b) .x .| by VALUED_1:18
   .= |. a * x + b .| by FCONT_1:def 4;
   per cases;
    suppose A6: x>=(-b)/a; then
     x*a >= (-b)/a*a by A1,XREAL_1:64; then
     x*a >= (-b) by A1,XCMPLX_1:87; then
     A8: a*x+b >= -b+b by XREAL_1:6;
     x in [.(-b)/a,+infty.[ by XXREAL_1:236,A6; then
     x in dom (AffineMap(a,b) | [.(-b)/a,+infty.[ ) by FUNCT_2:def 1;
     then
     ( (- (AffineMap(a,b) | ].-infty,(-b)/a.[ ))
       +* (AffineMap(a,b) | [.(-b)/a,+infty.[ )) . x
      = (AffineMap(a,b) | [.(-b)/a,+infty.[ ) . x by FUNCT_4:13
     .= AffineMap(a,b).x by FUNCT_1:49,XXREAL_1:236,A6
     .= a*x+b by FCONT_1:def 4;
     hence thesis by A5,A8,COMPLEX1:43;
    end;
    suppose B6: x <(-b)/a; then
     x*a < (-b)/a*a by A1,XREAL_1:68; then
     x*a < (-b) by A1,XCMPLX_1:87; then
     A8: a*x+b < -b+b by XREAL_1:6;
     not (x in dom (AffineMap(a,b) | [.(-b)/a,+infty.[ )) by B6,XXREAL_1:236;
     then
     ( (- (AffineMap(a,b) | ].-infty,(-b)/a.[ ))
       +* (AffineMap(a,b) | [.(-b)/a,+infty.[ )) . x
      = (- (AffineMap(a,b) | ].-infty,(-b)/a.[) ) . x by FUNCT_4:11
     .= - ((AffineMap(a,b) | ].-infty,(-b)/a.[).x) by VALUED_1:8
     .= -AffineMap(a,b).x by FUNCT_1:49,XXREAL_1:233,B6
     .= -(a*x+b) by FCONT_1:def 4;
     hence thesis by A5,A8,COMPLEX1:70;
    end;
  end;
  hence thesis by FUNCT_1:2,D1a,FUNCT_2:52;
end;
