reserve A for non empty closed_interval Subset of REAL;

theorem L724p:
for a,b,c,d,e be Real, f be Function of REAL,REAL st
b <> 0 & c <> 0 &
for x be Real holds f.x = min(d, max(e, b - |. b*(x-a)/c .|))
holds f is Lipschitzian
proof
 let a,b,c,d,e be Real, f be Function of REAL,REAL;
 assume A2:b <> 0 & c <> 0;
 assume A1:for x be Real holds f.x = min(d, max(e, b - |. b*(x-a)/c .|));
 ex r being Real st
 ( 0 < r &
 ( for x1, x2 being Real st x1 in dom f & x2 in dom f holds
  |.f.x1 - f.x2.| <= r * |.x1 - x2.| ) )
 proof
  take |.b.| * |. 1/c .|;
  TTT: |.b.| > 0 & |. (1/c) .| > 0 by A2,COMPLEX1:47;
  for x1, x2 being Real st x1 in dom f & x2 in dom f holds
  |.f.x1 - f.x2.| <= (|.b.| * |. 1/c .|) * |.x1 - x2.|
  proof
   let x1, x2 be Real;
   T1: |.f.x1 - f.x2.| = |.min(d, max(e, b - |. b*(x1-a)/c .|))
     - (f.x2).| by A1
     .=|.min(d, max(e, b - |. b*(x1-a)/c .|))
     - min(d, max(e, b - |. b*(x2-a)/c .|)).| by A1;
   T2: |.f.x1 - f.x2.|
    <= |.(b - |. b*(x1-a)/c .|) - (b - |. b*(x2-a)/c .|).| by T1,F51;
T41:   |. |. b*(x2-a)/c .|- |. b*(x1-a)/c .| .|
   <= |. b*(x2-a)/c - b*(x1-a)/c .| by COMPLEX1:64;
   |. b*(x2-a)/c - b*(x1-a)/c .|
      = |. b*((x2-a)/c) - b*(x1-a)/c .| by XCMPLX_1:74
     .= |. b*((x2-a)/c) - b*((x1-a)/c) .| by XCMPLX_1:74
     .= |. b*((x2-a)/c - (x1-a)/c) .|
     .= |.b.| * |. (x2-a)/c - (x1-a)/c .| by COMPLEX1:65
     .= |.b.| * |. (x2-a)*(1/c) - (x1-a)/c .| by XCMPLX_1:99
     .= |.b.| * |. (x2-a)*(1/c) - (x1-a)*(1/c) .| by XCMPLX_1:99
     .= |.b.| * |. (1/c)*((x2-a) - (x1-a)) .|
     .= |.b.| * (|. (1/c) .| * |.  ((x2-a) - (x1-a)) .| ) by COMPLEX1:65
     .= |.b.| * |. (1/c) .| * |.  x2 - x1 .|
     .= |.b.| * |. (1/c) .| * |.  x1 - x2 .| by COMPLEX1:60;
   hence thesis by T41,T2,XXREAL_0:2;
  end;
  hence thesis by TTT;
 end;
 hence thesis;
end;
