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
  AS is left_transitive implies AS is bach_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 & w
  ,w1 '//' v,v1 & w,w1 '//' u2,v2 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;
A2: AS is left_transitive implies AS is right_transitive by Th15;
    then
A3: u,u1 '//' v,v1 & v,v1 '//' w1,w & u2,v2 '//' w1,w implies thesis by A1;
    assume that
A4: u,u1 '//' v,v1 and
A5: w,w1 '//' v,v1 and
A6: w,w1 '//' u2,v2;
A7: ( v=v1 or w=w1 or u,u1 '//' u2,v2 or u,u1 '//' v2,u2) implies (w=w1 or
    v=v1 or u,u1 '//' u2,v2)
    proof
A8:   now
        assume that
A9:     u,u1 '//' v2,u2 and
A10:    u<>u1 and
A11:    v<>v1 and
        w<>w1;
A12:    now
          assume
A13:      u2,v2 '//' u,u1;
          then
A14:      u2,v2 '//' w,w1 by A2,A1,A4,A5,A10,A11;
A15:      now
            assume u2,v2 '//' w1,w;
            then w=w1 or u2=v2 by A14,Def1;
            hence thesis;
          end;
          w1,w '//' v2,u2 by A6,Def1;
          then u2,v2 '//' w1,w or u2=v2 by A2,A1,A9,A10,A13;
          hence thesis by A15;
        end;
A16:    now
A17:      w1,w '//' v1,v by A5,Def1;
          assume
A18:      u2,v2 '//' u1,u;
          u1,u '//' v1,v by A4,Def1;
          then
A19:      u2,v2 '//' w1,w by A2,A1,A10,A11,A18,A17;
A20:      now
            assume u2,v2 '//' w,w1;
            then w=w1 or u2=v2 by A19,Def1;
            hence thesis;
          end;
          u1,u '//' u2,v2 by A9,Def1;
          then u2,v2 '//' w,w1 or u2=v2 by A2,A1,A6,A10,A18;
          hence thesis by A20;
        end;
        v2,u2 '//' u,u1 or v2,u2 '//' u1,u by A9,Def1;
        hence thesis by A16,A12,Def1;
      end;
      assume v=v1 or w=w1 or u,u1 '//' u2,v2 or u,u1 '//' v2,u2;
      hence thesis by A8,Def1;
    end;
A21: now
      assume that
A22:  u,u1 '//' v,v1 and
A23:  v,v1 '//' w,w1 and
A24:  u2,v2 '//' w1,w;
      v2,u2 '//' w,w1 by A24,Def1;
      hence thesis by A2,A1,A7,A22,A23;
    end;
A25: now
      assume that
A26:  u,u1 '//' v,v1 and
A27:  v,v1 '//' w1,w and
A28:  u2,v2 '//' w,w1;
      v2,u2 '//' w1,w by A28,Def1;
      hence thesis by A2,A1,A7,A26,A27;
    end;
    u,u1 '//' v,v1 & v,v1 '//' w,w1 & u2,v2 '//' w,w1 implies thesis by A2,A1;
    hence thesis by A4,A5,A6,A21,A3,A25,Def1;
  end;
  hence thesis;
end;
