
theorem Th4:
  for n be non zero Element of NAT,
      x be Element of REAL-NS n,
      i be Element of NAT holds
   ||. Proj(i,n).x .|| = |. proj(i,n).x .|
proof
let n be non zero Element of NAT;
let x be Element of REAL-NS n;
let i be Element of NAT;
reconsider y = x as Element of REAL n by REAL_NS1:def 4;
Proj(i,n).x = <* proj(i,n).x *> by PDIFF_1:def 4
           .= <* y.i *> by PDIFF_1:def 1; then
||. Proj(i,n).x .|| = |.y.i.| by Th2;
hence thesis by PDIFF_1:def 1;
end;
