reserve E,F,G for RealNormSpace;
reserve f for Function of E,F;
reserve g for Function of F,G;
reserve a,b,c for Point of E;
reserve t for Real;

theorem Th15:
  f is Affine & g is Affine implies g*f is Affine
  proof
    assume that
A1: f is Affine and
A2: g is Affine;
    set h = g*f;
    let a,b,t such that
A3: 0 <= t & t <= 1;
A4: h.a = g.(f.a) & h.b = g.(f.b) by FUNCT_2:15;
    thus h.((1-t)*a+t*b) = g.(f.((1-t)*a+t*b)) by FUNCT_2:15
    .= g.((1-t)*f.a+t*f.b) by A1,A3
    .= (1-t)*(h.a) + t*(h.b) by A2,A3,A4;
  end;
