 reserve A for non empty Subset of REAL;

theorem Lm20A:
  for a,b,c be Real, f be Function of REAL,REAL st
    b > 0 & c > 0 &
  (for x be Real holds f.x = max(0, b - |. b*(x-a)/c .|)) holds
    for x be Real st not x in ['a-c,a+c'] holds f.x = 0
proof
 let a,b,c be Real;
 let f be Function of REAL,REAL;
 assume A1: b > 0 & c > 0; then
 B1: -c < 0;
 assume A2: for x be Real holds f.x = max(0,b-|. b*(x-a)/c .|);
 for x be Real st not x in ['a-c,a+c'] holds f.x = 0
 proof
  let x be Real;
  assume A4: not x in ['a-c,a+c'];
  a < a+c & a-c < a by XREAL_1:44, XREAL_1:29,A1; then
  A3: not x in [. a-c,a+c .] by INTEGRA5:def 3,A4,XXREAL_0:2;
  per cases by A3;
   suppose x < a-c; then
   B2: x-a < a-c-a by XREAL_1:9; then
   b*(x-a) < b*(-c) by A1,XREAL_1:68; then
   b*(x-a)/c < (-c*b)/c by A1,XREAL_1:74; then
   b*(x-a)/c < (-c*b/c) by XCMPLX_1:187; then
   b*(x-a)/c < -b by A1,XCMPLX_1:89; then
   B4: b*(x-a)/c + b < -b+b by XREAL_1:6;
   thus f.x = max(0, b - |. b*(x-a)/c .|) by A2
   .= max(0, b - -(b*(x-a)/c)) by COMPLEX1:70,A1,B1,B2
   .= 0 by B4,XXREAL_0:def 10;
   end;
   suppose a+c < x; then
   C2: x-a > a+c-a by XREAL_1:9; then
   b*(x-a) > b*c by A1,XREAL_1:68; then
   b*(x-a)/c > b*c/c by A1,XREAL_1:74; then
   b*(x-a)/c > b by A1,XCMPLX_1:89; then
   C4: b*(x-a)/c - b*(x-a)/c > b - b*(x-a)/c by XREAL_1:9;
   thus f.x = max(0, b - |. b*(x-a)/c .|) by A2
   .= max(0, b - (b*(x-a)/c)) by COMPLEX1:43,A1,C2
   .= 0 by C4,XXREAL_0:def 10;
   end;
 end;
 hence thesis;
end;
