reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th91:
  not 3 divides b implies not 3 divides b|^2 + c|^2
  proof
  L1: not 3 divides b & not 3 divides c implies not 3 divides b|^2 + c|^2
  proof
    assume
    A0: not 3 divides b & not 3 divides c;
    2+1 = 3; then
    3 divides b|^2 - 1 & 3 divides c|^2 - 1 by A0,Th59,PEPIN:41; then
    A1: 3 divides (b|^2 -1) + (c|^2 - 1) by WSIERP_1:4;
    b|^2 + c|^2 = (b|^2 + c|^2 - 2) + 2;
    hence thesis by A1,INT_2:1,27;
  end;
  not 3 divides b & 3 divides c implies not 3 divides b|^2 + c|^2
  proof
    assume
    A1: not 3 divides b & 3 divides c;
    c divides c|^2 by NAT_3:3; then
    not 3 divides b|^2 & 3 divides c|^2 by INT_2:9,A1,NAT_3:5,PEPIN:41;
    hence thesis by INT_2:1;
  end;
  hence thesis by L1;
end;
