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 & i*j mod p = 1 implies a" = j mod p
  proof
    assume A1: a = i mod p & i*j mod p = 1;
    reconsider b = j mod p as Element of GF(p) by Th14;
A2: p > 1 by INT_2:def 4;
A3: b*a = 1 by A1,Th18
    .= 1.GF(p) by A2,INT_3:14; then
    a <> 0.(GF(p));
    hence thesis by A3,VECTSP_1:def 10;
  end;
