
theorem
  for A being non-empty partial UAStr, i,j being Element of NAT,
  R being Relation of the carrier of A holds R|^(A,i+j) = R|^(A,i)|^(A,j)
proof
  let A be non-empty partial UAStr;
  let i,j be Element of NAT;
  let R be Relation of the carrier of A;
  defpred P[Nat] means R|^(A,i+$1) = R|^(A,i)|^(A,$1);
A1: P[ 0 ] by Th15;
A2: now
    let j be Nat;
    assume
A3: P[j];
    R|^(A,i+(j+1)) = R|^(A,(i+j)+1) .= R|^(A,i+j)|^A by Th16
      .= R|^(A,i)|^(A,j+1) by A3,Th16;
    hence P[j+1];
  end;
  for j being Nat holds P[j] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
