reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem Th32:
  for f,g being PartFunc of REAL,REAL, A being non empty closed_interval
  Subset of REAL st (f(#)f)||A is total & (f(#)g)||A is total & (g(#)g)||A is
  total & (f(#)f)||A|A is bounded & (f(#)g)||A|A is bounded & (g(#)g)||A|A is
  bounded & (f(#)f) is_integrable_on A & (f(#)g) is_integrable_on A & (g(#)g)
is_integrable_on A & f is_orthogonal_with g,A holds |||(f+g,f+g,A)||| = |||(f,f
  ,A)||| + |||(g,g,A)|||
proof
  let f,g be PartFunc of REAL,REAL;
  let A be non empty closed_interval Subset of REAL;
  assume (f(#)f)||A is total & (f(#)g)||A is total & (g(#)g)||A is total & (f
(#)f ) ||A|A is bounded & (f(#)g)||A|A is bounded & (g(#)g)||A|A is bounded & (
  f(#)f ) is_integrable_on A & (f(#)g) is_integrable_on A & (g(#)g)
  is_integrable_on A;
  then
A1: |||(f+g,f+g,A)||| = |||(f,f,A)||| + 2*|||(f,g,A)||| + |||(g,g,A)||| by Th31
;
  assume f is_orthogonal_with g,A;
  then |||(f,g,A)||| = 0;
  hence thesis by A1;
end;
