reserve a,b,c,d for Real;
reserve r,s for Real;

theorem Th167:
  (for a st 1 < a holds c <= b*a) implies c <= b
proof
  assume
A1: for a st a>1 holds b*a>=c;
  assume
A2: not thesis;
A3: b>=0
  proof
A4: c <= b*2 by A1;
    assume b<0;
    then
A5: b*2<0;
    then c < 0 by A1;
    then b/c>c/c by A2,Lm29;
    then b/c>1 by A5,A4,XCMPLX_1:60;
    then
A6: b*(b/c)>=c by A1;
    b*(b/c)<c*(b/c) by A2,A5,A4,Lm13;
    then b*(b/c)<b by A5,A4,XCMPLX_1:87;
    hence contradiction by A2,A6,XXREAL_0:2;
  end;
  per cases by A3;
  suppose
A7: b>0;
    then b/b<c/b by A2,Lm26;
    then 1<c/b by A7,XCMPLX_1:60;
    then consider d being Real such that
A8: 1<d and
A9: d<c/b by Th5;
    b*d<b*(c/b) by A7,A9,Lm13;
    then b*d<c by A7,XCMPLX_1:87;
    hence contradiction by A1,A8;
  end;
  suppose
A10: b=0;
    b*2>=c by A1;
    hence contradiction by A2,A10;
  end;
end;
