reserve V for RealLinearSpace;
reserve x,y for VECTOR of V;
reserve AS for Oriented_Orthogonality_Space;
reserve u,u1,u2,u3,v,v1,v2,v3,w,w1 for Element of AS;

theorem Th15:
  AS is left_transitive implies AS is right_transitive
proof
  (for u,u1,u2,v,v1,v2,w,w1 being Element of AS holds ( (u,u1 '//' v,v1 &
v,v1 '//' w,w1 & u,u1 '//' u2,v2 ) implies (u=u1 or v=v1 or u2,v2 '//' w,w1 )))
implies for u,u1,u2,v,v1,v2,w,w1 being Element of AS holds ( u,u1 '//' v,v1 & v
  ,v1 '//' w,w1 & u2,v2 '//' w,w1 implies (w=w1 or v=v1 or u,u1 '//' u2,v2 ))
  proof
    assume
A1: for u,u1,u2,v,v1,v2,w,w1 being Element of AS holds ( u,u1 '//' v,
v1 & v,v1 '//' w,w1 & u,u1 '//' u2,v2 implies (u=u1 or v=v1 or u2,v2 '//' w,w1
    ));
    let u,u1,u2,v,v1,v2,w,w1;
    assume that
A2: u,u1 '//' v,v1 and
A3: v,v1 '//' w,w1 and
A4: u2,v2 '//' w,w1;
A5: ( w=w1 or v=v1 or u,u1 '//' u2,v2 or u,u1 '//' v2,u2 ) implies (w=w1
    or v=v1 or u,u1 '//' u2,v2 )
    proof
      now
A6:     now
          assume v2,u2 '//' w,w1;
          then u2,v2 '//' w1,w by Def1;
          then u2=v2 or w=w1 by A4,Def1;
          hence thesis;
        end;
        u=u1 implies thesis by Def1;
        hence thesis by A1,A2,A3,A6;
      end;
      hence thesis;
    end;
A7: now
      assume that
A8:   w,w1 '//' v,v1 and
A9:   v,v1 '//' u,u1 and
A10:  w,w1 '//' v2,u2;
      w=w1 or v=v1 or v2,u2 '//' u,u1 by A1,A8,A9,A10;
      hence thesis by A5,Def1;
    end;
A11: now
      assume that
A12:  w,w1 '//' v1,v and
A13:  v,v1 '//' u,u1 and
A14:  w,w1 '//' v2,u2;
      v1,v '//' u1,u by A13,Def1;
      then w=w1 or v=v1 or v2,u2 '//' u1,u by A1,A12,A14;
      then w=w1 or v=v1 or u2,v2 '//' u,u1 by Def1;
      hence thesis by A5,Def1;
    end;
A15: now
      assume that
A16:  w,w1 '//' v1,v and
A17:  v,v1 '//' u,u1 and
A18:  w,w1 '//' u2,v2;
      v1,v '//' u1,u by A17,Def1;
      then w=w1 or v=v1 or u2,v2 '//' u1,u by A1,A16,A18;
      then w=w1 or v=v1 or v2,u2 '//' u,u1 by Def1;
      hence thesis by A5,Def1;
    end;
A19: now
      assume that
A20:  w,w1 '//' v1,v and
A21:  v,v1 '//' u1,u and
A22:  w,w1 '//' v2,u2;
      v1,v '//' u,u1 by A21,Def1;
      then w=w1 or v=v1 or v2,u2 '//' u,u1 by A1,A20,A22;
      hence thesis by A5,Def1;
    end;
A23: now
      assume that
A24:  w,w1 '//' v1,v and
A25:  v,v1 '//' u1,u and
A26:  w,w1 '//' u2,v2;
      v1,v '//' u,u1 by A25,Def1;
      then w=w1 or v1=v or u2,v2 '//' u,u1 by A1,A24,A26;
      hence thesis by A5,Def1;
    end;
A27: now
      assume that
A28:  w,w1 '//' v,v1 and
A29:  v,v1 '//' u1,u and
A30:  w,w1 '//' v2,u2;
      w=w1 or v=v1 or v2,u2 '//' u1,u by A1,A28,A29,A30;
      then w=w1 or v=v1 or u1,u '//' u2,v2 or u1,u '//' v2,u2 by Def1;
      hence thesis by A5,Def1;
    end;
A31: now
      assume that
A32:  w,w1 '//' v,v1 and
A33:  v,v1 '//' u1,u and
A34:  w,w1 '//' u2,v2;
      w=w1 or v=v1 or u2,v2 '//' u1,u by A1,A32,A33,A34;
      then w=w1 or v=v1 or u1,u '//' u2,v2 or u1,u '//' v2,u2 by Def1;
      hence thesis by A5,Def1;
    end;
    now
      assume that
A35:  w,w1 '//' v,v1 and
A36:  v,v1 '//' u,u1 and
A37:  w,w1 '//' u2,v2;
      w=w1 or v=v1 or u2,v2 '//' u,u1 by A1,A35,A36,A37;
      hence thesis by A5,Def1;
    end;
    hence thesis by A2,A3,A4,A7,A31,A27,A15,A11,A23,A19,Def1;
  end;
  hence thesis;
end;
