reserve a,b,c,d,e for Real;

theorem Th04:
  (for c st 0 < c holds |.a - b.| <= c) implies a = b
  proof
    assume
A1: for c st 0 < c holds |.a - b.| <= c;
    assume a <> b;
    then
A2: a - b <> 0;
    set e = |.a - b.| / 2;
    2 * e <= e by A1;
    then (2 * e) / e <= e / e by XREAL_1:72;
    then 2 <= e / e by A2,XCMPLX_1:89;
    then 2 <= 1 by A2,XCMPLX_1:60;
    hence contradiction;
  end;
