let V be RealLinearSpace; :: thesis: for u, u1, v, v1, x, y being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrte_wrt x,y holds
v,v1,u1,u are_COrte_wrt x,y
let u, u1, v, v1, x, y be VECTOR of V; :: thesis: ( Gen x,y & u,u1,v,v1 are_COrte_wrt x,y implies v,v1,u1,u are_COrte_wrt x,y )
assume A1: Gen x,y ; :: thesis: ( not u,u1,v,v1 are_COrte_wrt x,y or v,v1,u1,u are_COrte_wrt x,y )
assume u,u1,v,v1 are_COrte_wrt x,y ; :: thesis: v,v1,u1,u are_COrte_wrt x,y
then Orte (x,y,u), Orte (x,y,u1) // v,v1 ;
then v,v1 // Orte (x,y,u), Orte (x,y,u1) by ANALOAF:12;
then Orte (x,y,v), Orte (x,y,v1) // Orte (x,y,(Orte (x,y,u))), Orte (x,y,(Orte (x,y,u1))) by A1, Th16;
then Orte (x,y,v), Orte (x,y,v1) // - u, Orte (x,y,(Orte (x,y,u1))) by A1, Th14;
then Orte (x,y,v), Orte (x,y,v1) // - u, - u1 by A1, Th14;
then A2: - u, - u1 // Orte (x,y,v), Orte (x,y,v1) by ANALOAF:12;
(- u1) - (- u) = u + (- u1) by RLVECT_1:17
.= u - u1 ;
then A3: - u, - u1 // u1,u by ANALOAF:15;
A4: ( - u <> - u1 implies v,v1,u1,u are_COrte_wrt x,y ) by A2, A3, ANALOAF:11;
now :: thesis: ( - u = - u1 implies v,v1,u1,u are_COrte_wrt x,y )
assume - u = - u1 ; :: thesis: v,v1,u1,u are_COrte_wrt x,y
then u = - (- u1) by RLVECT_1:17
.= u1 by RLVECT_1:17 ;
then Orte (x,y,v), Orte (x,y,v1) // u1,u by ANALOAF:9;
hence v,v1,u1,u are_COrte_wrt x,y ; :: thesis: verum
end;
hence v,v1,u1,u are_COrte_wrt x,y by A4; :: thesis: verum