
theorem Th2:
  for a, b, c being Real st a > 0 & b >= 0 & c >= 0 & b < a
  & c < a for i being Integer st b = c + a*i holds b = c
proof
  let a, b, c be Real such that
A1: a > 0 and
A2: b >= 0 and
A3: c >= 0 and
A4: b < a and
A5: c < a;
A6: 0+a <= c+a by A3,XREAL_1:7;
  let i be Integer such that
A7: b = c + a*i;
  per cases;
  suppose
    i < 0;
    then i <= -1 by INT_1:8;
    then a*i <= a*(-1) by A1,XREAL_1:64;
    then c+a*i <= c-a by XREAL_1:7;
    hence thesis by A2,A5,A7,XREAL_1:49;
  end;
  suppose
    i = 0;
    hence thesis by A7;
  end;
  suppose
    i > 0;
    then 0+1 <= i by INT_1:7;
    then a*i >= a by A1,XREAL_1:151;
    then c+a*i >= c+a by XREAL_1:7;
    hence thesis by A4,A7,A6,XXREAL_0:2;
  end;
end;
