
theorem Th7:
  for x,p,y be Element of INT
  st y = x mod p & ALGO_EXGCD(p,y)`3_3 = 1
  holds
  ( ALGO_INVERSE(x,p) * x ) mod p = 1 mod p
  proof
    let x,p,y be Element of INT;
    assume A1: y = x mod p & ALGO_EXGCD(p,y)`3_3 = 1;
    per cases;
    suppose A2: p = 0;
      hence ( ALGO_INVERSE(x,p) * x ) mod p = 0 by INT_1:def 10
      .= 1 mod p by A2,INT_1:def 10;
    end;
    suppose p <> 0;
      hence ( ALGO_INVERSE(x,p) * x ) mod p = 1 mod p by Lm13,A1;
    end;
  end;
