
theorem
  for a,b,c being Nat st
    a ^2 + b ^2 = c ^2 &
    a,b,c form_an_AP holds
   ex i being Integer st
    a = 3 * i & b = 4 * i & c = 5 * i
  proof
    let a,b,c be Nat;
    assume
A1: a ^2 + b ^2 = c ^2 &
    a,b,c form_an_AP;
    set r = b - a;
    per cases;
    suppose
A2:   b <> 0;
      (b - r) ^2 + b ^2 = (b + r) ^2 by A1; then
      b = 4 * r * b / b by A2,XCMPLX_1:89; then
A4:   b = 4 * r by A2,XCMPLX_1:89; then
B1:   a = 3 * r;
B2:   c = 5 * r by A4,A1;
      reconsider rr = r as Integer;
      take rr;
      thus thesis by B1,B2;
    end;
    suppose
C1:   b = 0;
      take i = 0;
      thus thesis by C1,A1;
    end;
  end;
