theorem Th14:
  w is_atlas_of S,G implies for a,b,b9,c,c9 holds w.(a,b) = w.(b,c
  ) & w.(a,b9) = w.(b9,c9) implies w.(c,c9) = Double w.(b,b9)
proof
  assume
A1: w is_atlas_of S,G;
  let a,b,b9,c,c9;
  assume
A2: w.(a,b) = w.(b,c) & w.(a,b9) = w.(b9,c9);
  thus w.(c,c9) = w.(c,b9) + w.(b9,c9) by A1
    .= w.(c,a) + w.(a,b9) + w.(b9,c9) by A1
    .= w.(c,b) + w.(b,a) + w.(a,b9) + w.(b9,c9) by A1
    .= Double w.(b,a) + w.(a,b9) + w.(a,b9) by A1,A2,Th5
    .= Double w.(b,a) + Double w.(a,b9) by RLVECT_1:def 3
    .= Double (w.(b,a) + w.(a,b9)) by Th10
    .= Double w.(b,b9) by A1;
end;
