
theorem LM6A:
  for X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL
  st f=g holds
   f is additive homogeneous iff g is additive homogeneous
proof
  let X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL;
  assume AS: f=g;
  set Y=RUSp2RNSp X;
  hereby assume AS1: f is additive homogeneous;
A1: g is additive
    proof
      let x,y be Point of Y;
      reconsider x1=x,y1=y as Point of X;
      thus g.(x+y) = f.(x1+y1) by AS
                  .= g.x + g.y by AS,AS1,HAHNBAN:def 2;
    end;
    g is homogeneous
    proof
      let x be Point of Y, r be Real;
      reconsider x1=x as Point of X;
      thus g.(r*x) = f.(r*x1) by AS
                  .= r*g.x by AS,AS1,HAHNBAN:def 3;
   end;
   hence g is additive homogeneous by A1;
  end;
  assume AS2: g is additive homogeneous;
A2: f is additive
  proof
    let x,y be Point of X;
    reconsider x1=x,y1=y as Point of Y;
    thus f.(x+y) = g.(x1+y1) by AS
                .= f.x + f.y by AS,AS2,HAHNBAN:def 2;
  end;
  f is homogeneous
  proof
    let x be Point of X, r be Real;
    reconsider x1=x as Point of Y;
    thus f.(r*x) = g.(r*x1) by AS
                .= r*f.x by AS,AS2,HAHNBAN:def 3;
  end;
  hence f is additive homogeneous by A2;
end;
