reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;

theorem Th21:
  x,y '||' x,z implies y,x '||' y,z
proof
A1: now
    assume x,y // z,x;
    then y,x // x,z by ANALOAF:def 5;
    then y,x // y,z by ANALOAF:def 5;
    hence thesis;
  end;
A2: now
A3: now
      assume x,z // z,y;
      then y,z // z,x by Th2;
      then y,z // y,x by ANALOAF:def 5;
      then y,x // y,z by Th2;
      hence thesis;
    end;
A4: x,y // y,z implies thesis by ANALOAF:def 5;
    assume x,y // x,z;
    hence thesis by A4,A3,ANALOAF:def 5;
  end;
  assume x,y '||' x,z;
  hence thesis by A2,A1;
end;
