theorem
  ex c st PSym(a,PSym(c,p)) = PSym(c,PSym(b,p))
proof
  consider c such that
A1: Mid a,c,b by Th19;
  PSym(b,p) = PSym(PSym(c,a),p) by A1,Def4
    .= PSym(c,PSym(a,(PSym(c,p)))) by Th37;
  then PSym(c,PSym(b,p)) = PSym(a,(PSym(c,p))) by Th29;
  hence thesis;
end;
