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 Th13:
  f is midpoints-preserving & g is midpoints-preserving implies
  g*f is midpoints-preserving
  proof
    assume that
A1: f is midpoints-preserving and
A2: g is midpoints-preserving;
    set h = g*f;
    let a,b;
A3: h.a = g.(f.a) & h.b = g.(f.b) by FUNCT_2:15;
    thus h.(1/2*(a+b)) = g.(f.(1/2*(a+b))) by FUNCT_2:15
    .= g.(1/2*((f.a)+(f.b))) by A1
    .= 1/2*(h.a+h.b) by A3,A2;
  end;
