reserve Z for set;

theorem
  for n be Element of NAT, A be non empty closed_interval Subset of REAL, f be
  PartFunc of REAL,REAL n, g be Function of A,REAL n st f|A = g holds
  integral(f,A) = integral(g)
  proof
  let n be Element of NAT, A be non empty closed_interval Subset of REAL,
  f be PartFunc
  of REAL,REAL n, g be Function of A,REAL n;
  assume
A1: f|A = g;
A2: now
    let k be Nat;
    assume
A3: k in dom (integral(f,A));
    then reconsider i=k as Element of NAT;
    dom (proj(i,n))=REAL n by FUNCT_2:def 1;
    then rng f c= dom(proj(i,n));
    then
A4: dom(proj(i,n)*f) = dom f by RELAT_1:27;
    A = dom g by FUNCT_2:def 1
      .= dom f /\ A by A1,RELAT_1:61;
    then (proj(i,n)*f)||A is total by A4,INTEGRA5:6,XBOOLE_1:17;
    then reconsider F = (proj(i,n)*f)|A as Function of A,REAL;
A5: F = proj(i,n)*g by A1,Lm6;
A6: i in Seg n by A3,Def17;
    then integral(f,A).i = integral((proj(i,n)*f), A) by Def17
      .= integral((proj(i,n)*f)||A );
    hence integral(f,A).k = (integral(g)).k by A6,A5,Def14;
  end;
  dom (integral(f,A)) = Seg n by Def17
    .= dom (integral(g)) by Def14;
  hence thesis by A2,FINSEQ_1:13;
end;
