reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem
  p > 1 implies order(1,p) = 1
proof
  assume
A1: p > 1;
  p gcd 1 = 1 by NEWTON:51;
  then
A2: 1,p are_coprime by INT_2:def 3;
  (1 |^ 1) mod p = 1 by A1,NAT_D:24;
  then order(1,p) <= 1 by A1,A2,Def2;
  then order(1,p) < 1 or order(1,p) = 1 by XXREAL_0:1;
  then order(1,p) = 0 or order(1,p) = 1 by NAT_1:14;
  hence thesis by A1,A2,Def2;
end;
