reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;

theorem Th35:
  for p being FinSequence of REAL
  for M being Matrix of 3,REAL st len p = 3 holds
  SumAll QuadraticForm (a * p,M,b * p) = a * b * SumAll QuadraticForm(p,M,p)
  proof
    let p be FinSequence of REAL;
    let M be Matrix of 3,REAL;
    assume
A1: len p = 3;
A2: len M = 3 & width M = 3 by MATRIX_0:23; then
A3: len (a * p) = len M by A1,RVSUM_1:117;
    len (b * p) = width M & len (b * p) > 0 by A1,A2,RVSUM_1:117; then
A4: SumAll QuadraticForm (a * p,M,b * p) = |( a * p, M * (b * p) )|
      by A3,MATRPROB:44;
    len p = len M & len p = width M & len p > 0 by A1,MATRIX_0:23;
    then SumAll QuadraticForm(p,M,p) = |( p, M * p )| by MATRPROB:44;
    hence thesis by A4,A1,Th34;
  end;
