 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 A2: a < 0;
  set f = ( (AffineMap(a,b) | ].-infty,(-b)/a.[ ))
       +* - (AffineMap(a,b) | [.(-b)/a,+infty.[ );
  A1: dom(-(AffineMap(a,b) | [.(-b)/a,+infty.[) ) = [.(-b)/a,+infty.[ &
  [.(-b)/a,+infty.[ = dom((- AffineMap(a,b)) | [.(-b)/a,+infty.[)
   by FUNCT_2:def 1;
  for x being object st x in dom ((- AffineMap(a,b)) | [.(-b)/a,+infty.[)
  holds
  (- (AffineMap(a,b) | [.(-b)/a,+infty.[)).x
  = ((- AffineMap(a,b)) | [.(-b)/a,+infty.[).x
  proof
   let x be object;
   assume
   A3: x in dom ((- AffineMap(a,b)) | [.(-b)/a,+infty.[);
   hence ((- AffineMap(a,b)) | [.(-b)/a,+infty.[).x
    = (- AffineMap(a,b)).x by FUNCT_1:49
   .=- (AffineMap(a,b).x) by VALUED_1:8
   .=- (AffineMap(a,b) | [.(-b)/a,+infty.[.x) by FUNCT_1:49,A3
   .=(- (AffineMap(a,b) | [.(-b)/a,+infty.[)).x by VALUED_1:8;
  end; then
  - (AffineMap(a,b) | [.(-b)/a,+infty.[)
  = (- AffineMap(a,b)) | [.(-b)/a,+infty.[ by FUNCT_1:2,A1; then
  reconsider f as Function of REAL,REAL by FUZZY_6:18;
  for x being Element of REAL holds f . x = |. AffineMap (a,b) .| . x
  proof
   let x be Element of REAL;
   per cases;
    suppose A6: x>=(-b)/a; then
     x*a <= (-b)/a*a by A2,XREAL_1:65; then
     x*a <= (-b) by A2,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;
     hence
     f.x = ( -( AffineMap(a,b) | [.(-b)/a,+infty.[ ) ).x by FUNCT_4:13
        .= ( -( AffineMap(a,b) | [.(-b)/a,+infty.[ ).x ) by VALUED_1:8
        .= -( AffineMap(a,b).x ) by FUNCT_1:49,XXREAL_1:236,A6
        .= -(a*x+b) by FCONT_1:def 4
        .= |. a*x+b .| by COMPLEX1:70,A8
        .= |. AffineMap (a,b) . x .| by FCONT_1:def 4
        .= |. AffineMap (a,b) .| . x by VALUED_1:18;
    end;
    suppose B5: x<(-b)/a; then
     x*a > (-b)/a*a by A2,XREAL_1:69; then
     x*a > (-b) by A2,XCMPLX_1:87; then
     B8: a*x+b > -b+b by XREAL_1:6;
     not x in dom (-( AffineMap(a,b) | [.(-b)/a,+infty.[ ))
         by B5,XXREAL_1:236;
     hence f.x = (AffineMap(a,b) | ].-infty,(-b)/a.[ ).x by FUNCT_4:11
     .= AffineMap(a,b).x by FUNCT_1:49,XXREAL_1:233,B5
     .= a*x+b by FCONT_1:def 4
     .= |. a*x+b .| by COMPLEX1:43,B8
     .= |. AffineMap (a,b) . x .| by FCONT_1:def 4
     .= |. AffineMap (a,b) .| . x by VALUED_1:18;
    end;
  end;
  hence thesis by FUNCT_2:63;
end;
