 reserve A for non empty Subset of REAL;

theorem Th1:
  for a,b,c be Real st b > 0 & c > 0 holds
  for x be Real holds
    ( AffineMap ( b/c,b-a*b/c) | ]. -infty,a .]
    +* AffineMap (-b/c,b+a*b/c) | [. a,+infty .[ ).x =
      b - |. b*(x-a)/c .|
proof
 let a,b,c be Real;
 assume A1: b > 0;
 assume A2: c > 0;
 for x being Real holds
( AffineMap ( b/c,b-a*b/c) | ]. -infty,a .]
+* AffineMap (-b/c,b+a*b/c) | [. a,+infty .[ ).x = b -|. b*(x-a)/c .|
 proof
  let x be Real;
  D2: dom ( AffineMap (-b/c,b+a*b/c) | [. a,+infty .[)
                                     = [. a,+infty .[ by FUNCT_2:def 1;
  per cases;
  suppose A3: x<a; then
   A31: a-x >x-x by XREAL_1:9;
   A32: ( AffineMap ( b/c,b-a*b/c) | ]. -infty,a .]
   +* AffineMap (-b/c,b+a*b/c) | [. a,+infty .[ ).x
    = ( AffineMap ( b/c,b-a*b/c) | ]. -infty,a .] ).x
              by FUNCT_4:11,D2,XXREAL_1:236,A3
   .= (AffineMap ( b/c,b-a*b/c)).x by FUNCT_1:49,A3,XXREAL_1:234
   .= b/c*x + (b-a*b/c) by FCONT_1:def 4
   .= x*b/c + (b-a*b/c) by XCMPLX_1:74;
   b-|. b*(x-a)/c .| = b-|. b*(x-a) .|/|. c .| by COMPLEX1:67
   .= b-|. b .|*|. (x-a) .|/|. c .| by COMPLEX1:65
   .= b- b *|. (x-a) .|/|. c .| by COMPLEX1:43,A1
   .= b- b *|. (x-a) .|/ c by COMPLEX1:43,A2
   .= b- b *|. (a-x) .|/ c by COMPLEX1:60
   .= b- b * (a-x) / c by COMPLEX1:43,A31
   .= b - (b*a - b*x)/c
   .= b - (b*a/c - b*x/c) by XCMPLX_1:120;
   hence thesis by A32;
  end;
  suppose A4:x>=a; then
   A41: x-a >=a-a by XREAL_1:9;
   A42: ( AffineMap ( b/c,b-a*b/c) | ]. -infty,a .]
   +* AffineMap (-b/c,b+a*b/c) | [. a,+infty .[ ).x
    = ( AffineMap (-b/c,b+a*b/c) | [. a,+infty .[ ).x
              by FUNCT_4:13,D2,XXREAL_1:236,A4
   .= (AffineMap ( -b/c,b+a*b/c)).x by FUNCT_1:49,A4,XXREAL_1:236
   .= (-b/c)*x + (b+a*b/c) by FCONT_1:def 4
   .= -(b/c*x) + (b+a*b/c)
   .= -(x*b/c) + (b+a*b/c) by XCMPLX_1:74;
   b-|. b*(x-a)/c .| = b-|. b*(x-a) .|/|. c .| by COMPLEX1:67
   .= b-|. b .|*|. (x-a) .|/|. c .| by COMPLEX1:65
   .= b- b *|. (x-a) .|/|. c .| by COMPLEX1:43,A1
   .= b- b *|. (x-a) .|/ c by COMPLEX1:43,A2
   .= b- b * (x-a) / c by COMPLEX1:43,A41
   .= b - (b*x - b*a)/c
   .= b - (b*x/c - b*a/c) by XCMPLX_1:120;
   hence thesis by A42;
  end;
 end;
 hence thesis;
end;
