reserve A for non empty closed_interval Subset of REAL;

theorem F51min:
  for a, b, c being Real holds
    |. min (c,a) - min (c,b) .| <= |. a - b .|
proof
 let a, b, c be Real;
 per cases;
  suppose A1:a <= b & b <= c; then
   |. min (c,a) - min (c,b) .| = |. a - min (c,b) .|
     by XXREAL_0:def 9,XXREAL_0:2;
   hence thesis by XXREAL_0:def 9,A1;
  end;
  suppose A2: a <= c & c <= b;
   then a <= b by XXREAL_0:2; then
  A22: a-a <= b-a by XREAL_1:13;
   A24: |. (a - b) .| = |. b-a .| by COMPLEX1:60
   .= b-a by ABSVALUE:def 1,A22;
 A21:  a-a <= c-a by A2,XREAL_1:13;
   |. min (c,a) - min (c,b) .| = |. a - min (c,b) .| by XXREAL_0:def 9,A2
   .= |. a - c .| by XXREAL_0:def 9,A2
   .= |. c-a .| by COMPLEX1:60
   .= c-a 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
   b-b <= a-b by XREAL_1:13; then
   A32: |. a - b .| = a - b by ABSVALUE:def 1;
A33:  b-b <= c-b by A3,XREAL_1:13;
   |. min (c,a) - min (c,b) .| = |. c - min (c,b) .| by XXREAL_0:def 9,A3
   .= |. c - b .| by XXREAL_0:def 9,A3
   .= c-b by ABSVALUE:def 1,A33;
   hence thesis by A32,A3,XREAL_1:13;
  end;
  suppose A4:b <= a & a <= c; then
   |. min (c,a) - min (c,b) .| = |. min (c,a) - b .|
      by XXREAL_0:def 9,XXREAL_0:2
   .= |. a - b .| by XXREAL_0:def 9,A4;
   hence thesis;
  end;
  suppose A5:c <= a & a <= b; then
   |. min (c,a) - min (c,b) .| = |. min (c,a) - c .|
      by XXREAL_0:def 9,XXREAL_0:2
    .= |. c - c .| by XXREAL_0:def 9,A5
    .= 0 by COMPLEX1:44;
   hence thesis by COMPLEX1:46;
  end;
  suppose A6:c <= b & b <= a; then
   |. min (c,a) - min (c,b) .| = |. c - min (c,b) .|
       by XXREAL_0:def 9,XXREAL_0:2
    .= |. c - c .| by XXREAL_0:def 9,A6
    .= 0 by COMPLEX1:44;
   hence thesis by COMPLEX1:46;
  end;
end;
