reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th9:
 for f1 be PartFunc of W,REAL-NS n,
     g1 be PartFunc of W,REAL n st f1=g1 holds
  ||. f1 .|| = |. g1 .|
proof
  let f1 be PartFunc of W,REAL-NS n,
      g1 be PartFunc of W,REAL n;
  assume A1: f1=g1;
  dom (||. f1 .||) = dom f1 by NORMSP_0:def 3; then
A2: dom (||. f1 .||) = dom(|. g1 .|) by A1,Def2;
  now
    let x be Element of W;
    assume A3: x in dom ||. f1 .||;
A4: f1/.x=g1/.x by A1,REAL_NS1:def 4;
    set y1=g1/.x;
A5: ||. f1/.x .||=|. y1 .| by A4,REAL_NS1:1;
A6: (||. f1 .||).x = ||. f1/.x .|| by A3,NORMSP_0:def 3;
    |. g1 .| /.x = |. g1/.x .| by A2,A3,Def2;
    hence (||. f1 .||).x = (|. g1 .|).x by A2,A3,A5,A6,PARTFUN1:def 6;
  end;
  hence thesis by A2,PARTFUN1:5;
end;
