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 Th14:
  f is bijective midpoints-preserving implies f/" is midpoints-preserving
  proof
    assume that
A1: f is bijective and
A2: f is midpoints-preserving;
    set g = f/";
    let a,b be Point of F;
A3: g*f = id E by A1,Lm3;
    f.(g.a) = a & f.(g.b) = b by A1,Lm2;
    hence g.(1/2*(a+b)) = g.(f.(1/2*((f/".a)+(f/".b)))) by A2
    .= (g*f).(1/2*((f/".a)+(f/".b))) by FUNCT_2:15
    .= 1/2*(g.a+g.b) by A3;
  end;
