reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th17:
  for f be PartFunc of REAL,REAL n, g be Function of A,REAL n
  st f = g holds |. f .| = |. g .|
  proof
    let f be PartFunc of REAL,REAL n, g be Function of A,REAL n;
    assume A1:f = g;
A2: dom (|. f .|) = dom f by NFCONT_4:def 2
    .= dom |. g .| by A1,NFCONT_4:def 2;
    now let x be object;
      assume A3: x in dom (|. f .|);
      thus (|. f .|).x  =|. f .| /.x by A3,PARTFUN1:def 6
      .=|. f/.x .| by A3,NFCONT_4:def 2
      .= (|. g .|)/.x by A1,A2,A3,NFCONT_4:def 2
      .= (|. g .|).x by A2,A3,PARTFUN1:def 6;
    end;
    hence thesis by A2,FUNCT_1:2;
  end;
