reserve A for non empty closed_interval Subset of REAL;

theorem F51max:
  for a, b, c being Real holds
    |. max (c,a) - max (c,b) .| <= |. a - b .|
proof
  let a, b, c be Real;
  per cases;
  suppose A1:a >= b & b >= c; then
   |. max (c,a) - max (c,b) .| = |. a - max (c,b) .|
     by XXREAL_0:def 10,XXREAL_0:2;
   hence thesis by XXREAL_0:def 10,A1;
  end;
  suppose A2:a >= c & c >= b;
   then a >= b by XXREAL_0:2; then
   a-b >= b-b by XREAL_1:13; then
   A24: |. (a - b) .| = a-b by ABSVALUE:def 1;
 A21:  a-c >= c-c by A2,XREAL_1:13;
   |. max (c,a) - max (c,b) .| = |. a - max (c,b) .| by XXREAL_0:def 10,A2
   .= |. a - c .| by XXREAL_0:def 10,A2
   .= a-c by ABSVALUE:def 1,A21;
   hence thesis by A24,A2,XREAL_1:13;
  end;
  suppose A3:b >= c & c >= a; then
   b >= a by XXREAL_0:2; then
 AA:  b-a >= a-a by XREAL_1:13;
   A32: |. (a - b) .| = |. b-a .| by COMPLEX1:60
   .= b-a by ABSVALUE:def 1,AA;
 A33:  b-c >= c-c by A3,XREAL_1:13;
   |. max (c,a) - max (c,b) .| = |. c - max (c,b) .| by XXREAL_0:def 10,A3
   .= |. c - b .| by XXREAL_0:def 10,A3
   .= |. b-c .| by COMPLEX1:60
   .= b-c by ABSVALUE:def 1,A33;
   hence thesis by A32,A3,XREAL_1:13;
  end;
  suppose A4:b >= a & a >= c; then
   |. max (c,a) - max (c,b) .| = |. max (c,a) - b .|
     by XXREAL_0:def 10,XXREAL_0:2
   .= |. a - b .| by XXREAL_0:def 10,A4;
   hence thesis;
  end;
  suppose A5:c >= a & a >= b; then
   |. max (c,a) - max (c,b) .| = |. max (c,a) - c .|
      by XXREAL_0:def 10,XXREAL_0:2
    .= |. c - c .| by XXREAL_0:def 10,A5
    .= 0 by COMPLEX1:44;
   hence thesis by COMPLEX1:46;
  end;
  suppose A6:c >= b & b >= a; then
   |. max (c,a) - max (c,b) .| = |. c - max (c,b) .|
      by XXREAL_0:def 10,XXREAL_0:2
    .= |. c - c .| by XXREAL_0:def 10,A6
    .= 0 by COMPLEX1:44;
   hence thesis by COMPLEX1:46;
  end;
end;
