reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;

theorem Th1:
  0 < i * r < r implies i = 1
  proof
    assume
A1: 0 < i * r < r;
    assume not i = 1;
    then 0 < 1 - i or 0 < i - 1 by XREAL_1:55;
    then 0 + i < 1 - i + i or 0 + 1 < i - 1 + 1 by XREAL_1:8;
    then per cases;
    suppose i < 1;
      then i <= 1 - 1 by INT_1:52;
      hence contradiction by A1;
    end;
    suppose i > 1;
      hence contradiction by A1,XREAL_1:155;
    end;
  end;
