let V be RealLinearSpace; :: thesis: for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y holds
( u,v,u1,v1 are_Ort_wrt x,y iff ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) )
let x, y, u, v, u1, v1 be VECTOR of V; :: thesis: ( Gen x,y implies ( u,v,u1,v1 are_Ort_wrt x,y iff ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) ) )
assume A1: Gen x,y ; :: thesis: ( u,v,u1,v1 are_Ort_wrt x,y iff ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) )

A2: now

assume u = v ; :: thesis: u,v,u1,v1 are_Ort_wrt x,y
then v - u = 0. V by RLVECT_1:28;
then v - u,v1 - u1 are_Ort_wrt x,y by A1, ANALMETR:9;
hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3; :: thesis: verum

end;

now

assume A3: u <> v ; :: thesis: ( u,v,u1,v1 are_Ort_wrt x,y iff ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) )
set u2 = Orte x,y,u;
set v2 = Orte x,y,v;
A4: v - u <> 0. V by A3, RLVECT_1:35;
u,v, Orte x,y,u, Orte x,y,v are_Ort_wrt x,y by A1, Th24;
then A5: v - u,(Orte x,y,v) - (Orte x,y,u) are_Ort_wrt x,y by ANALMETR:def 3;

A6: now

assume u,v,u1,v1 are_Ort_wrt x,y ; :: thesis: ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y )
then v - u,v1 - u1 are_Ort_wrt x,y by ANALMETR:def 3;
then ex a, b being Real st
( a * ((Orte x,y,v) - (Orte x,y,u)) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) by A1, A4, A5, ANALMETR:13;
then ( Orte x,y,u, Orte x,y,v // u1,v1 or Orte x,y,u, Orte x,y,v // v1,u1 ) by ANALMETR:18;
hence ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) by Def3; :: thesis: verum

end;

now

assume ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) ; :: thesis: u,v,u1,v1 are_Ort_wrt x,y
then ( Orte x,y,u, Orte x,y,v // u1,v1 or Orte x,y,u, Orte x,y,v // v1,u1 ) by Def3;
then consider a, b being Real such that
A7: a * ((Orte x,y,v) - (Orte x,y,u)) = b * (v1 - u1) and
A8: ( a <> 0 or b <> 0 ) by ANALMETR:18;

A9: now

assume A10: b = 0 ; :: thesis: u,v,u1,v1 are_Ort_wrt x,y
then 0. V = a * ((Orte x,y,v) - (Orte x,y,u)) by A7, RLVECT_1:23;
then (Orte x,y,v) - (Orte x,y,u) = 0. V by A8, A10, RLVECT_1:24;
then Orte x,y,v = Orte x,y,u by RLVECT_1:35;
then u = v by A1, Th13;
then v - u = 0. V by RLVECT_1:28;
then v - u,v1 - u1 are_Ort_wrt x,y by A1, ANALMETR:9;
hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3; :: thesis: verum

end;

now

assume A11: b <> 0 ; :: thesis: u,v,u1,v1 are_Ort_wrt x,y
((b " ) * a) * ((Orte x,y,v) - (Orte x,y,u)) = (b " ) * (b * (v1 - u1)) by A7, RLVECT_1:def 10;
then ((b " ) * a) * ((Orte x,y,v) - (Orte x,y,u)) = ((b " ) * b) * (v1 - u1) by RLVECT_1:def 10;
then ((b " ) * a) * ((Orte x,y,v) - (Orte x,y,u)) = 1 * (v1 - u1) by A11, XCMPLX_0:def 7;
then v1 - u1 = ((b " ) * a) * ((Orte x,y,v) - (Orte x,y,u)) by RLVECT_1:def 11;
then v - u,v1 - u1 are_Ort_wrt x,y by A5, ANALMETR:11;
hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3; :: thesis: verum

end;

hence u,v,u1,v1 are_Ort_wrt x,y by A9; :: thesis: verum

end;

hence ( u,v,u1,v1 are_Ort_wrt x,y iff ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) ) by A6; :: thesis: verum

end;

hence ( u,v,u1,v1 are_Ort_wrt x,y iff ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) ) by A2, Th20; :: thesis: verum