
theorem Th31:
  for a,b,c being Real st 0 <= a <= 1 & 0 < b * c holds
  0 <= (a * c) / ((1 - a) * b + a * c) <= 1
  proof
    let a,b,c be Real;
    assume that
A1: 0 <= a <= 1 and
A2: 0 < b * c;
    per cases by A2,XREAL_1:134;
    suppose
A3:   0 < b & 0 < c;
B3:   a - a <= 1 - a by A1,XREAL_1:9;
      hence 0 <= (a * c) / ((1 - a) * b + a * c) by A1,A3;
      0 + a * c <= (1 - a) * b + a * c by A3,B3,XREAL_1:6;
      hence (a * c) / ((1 - a) * b + a * c) <= 1 by A1,A3,XREAL_1:183;
    end;
    suppose
A4:   b < 0 & c < 0;
A5:   a - a <= 1 - a by A1,XREAL_1:9;
      hence 0 <= (a * c) / ((1 - a) * b + a * c) by A1,A4;
      (1 - a) * b + a * c <= 0 + a * c by A4,A5,XREAL_1:6;
      hence (a * c) / ((1 - a) * b + a * c) <= 1 by A1,A4,XREAL_1:184;
    end;
  end;
