reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve K,K1,K2,K3 for Field;
reserve SK1,SK2 for Subfield of K;
reserve ek,ek1,ek2 for Element of K;
reserve p for Prime;
reserve a,b,c for Element of GF(p);
reserve F for FinSequence of GF(p);

theorem
  a = i mod p & b = j mod p implies a-b = (i-j) mod p
  proof
    assume A1: a = i mod p & b = j mod p; then
    -b = (p-j) mod p by Th16;
    then a-b = (i+(p-j)) mod p by A1,Th15
    .= (i-j+1*p) mod p;
    hence thesis by NAT_D:61;
  end;
