reserve MS for non empty MidStr;
reserve a, b for Element of MS;
reserve M for MidSp;
reserve a,b,c,d,a9,b9,c9,d9,x,y,x9 for Element of M;
reserve p,q,r,p9,q9 for Element of [:the carrier of M,the carrier of M:];
reserve u,v,w,u9,w9 for Vector of M;

theorem Th46:
  u+v = v+u
proof
  set a = the Element of M;
  consider b such that
A1: u = vect(a,b) by Th35;
  consider c such that
A2: v = vect(b,c) by Th35;
  consider d such that
A3: v = vect(a,d) by Th35;
  consider c9 such that
A4: u = vect(d,c9) by Th35;
A5: a@c9 = b@d by A1,A4,Th29
    .= a@c by A2,A3,Th29;
  u + v = vect(a,c) by A1,A2,Th40
    .= vect(a,c9) by A5,Th8
    .= v + u by A3,A4,Th40;
  hence thesis;
end;
