
theorem LMINEQ:
  for x,y,z be Real st 0 <= y
   & for e be Real st 0 < e holds x <= z + y*e
  holds x <= z
  proof
    let x,y,z be Real such that
    A1: 0 <= y and
    A2: for e be Real st 0 < e holds x <= z + y*e;
    per cases;
    suppose
      A3: y = 0;
      x <= z + y*1 by A2;
      hence x <= z by A3;
    end;
    suppose
      A4: y<>0;
      thus x <= z
      proof
        assume not x <= z; then
        A6: 0 < x-z by XREAL_1:50; then
        x <= z + y*(((x-z)/2)/y) by A1,A2,A4; then
        A7: x <= z + (y*((x-z)/2))/y by XCMPLX_1:74;
        (x-z)/2 < x-z by A6,XREAL_1:216; then
        z + (x-z)/2 < z + (x-z) by XREAL_1:8;
        hence contradiction by A4,A7,XCMPLX_1:89;
      end;
    end;
  end;
