reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th11:
  p mod 4 = 3 implies p|^(2*n) mod 4 = 1 & p|^(2*n+1) mod 4 = 3
  proof
  assume
A1: p mod 4 = 3;
    defpred P[Nat] means p|^(2*$1) mod 4 = 1 & p|^(2*$1+1) mod 4 = 3;
    p|^(2*0) = 1 & p|^(2*0+1) = p by NEWTON:4;
    then
A2: P[0] by A1,PEPIN:5;
A3: for i being Nat holds P[i] implies P[i+1]
    proof
      let i be Nat;
      set i1 = i+1;
      assume
A4:   P[i];
A5:   p|^(2*i1+1) = p *  p|^(2*i+1+1) & p|^(2*i+1+1) = p* p|^(2*i+1)
      by NEWTON:6;
      then p|^(2*i+1+1) mod 4 = 1 by A4,A1,Th8;
      hence thesis by A5,A1,Th9;
    end;
    for i being Nat holds P[i] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
