let F be Field; :: thesis: for S being OrtSp of F
for a, b, c being Element of S st not a _|_ & a _|_ holds
not a _|_
let S be OrtSp of F; :: thesis: for a, b, c being Element of S st not a _|_ & a _|_ holds
not a _|_
let a, b, c be Element of S; :: thesis: ( not a _|_ & a _|_ implies not a _|_ )
assume that
A1: not a _|_ and
A2: a _|_ ; :: thesis: not a _|_
a _|_ by A2, Def1;
then A3: a _|_ by VECTSP_1:14;
assume a _|_ ; :: thesis: contradiction
then a _|_ by A3, Def1;
then a _|_ by RLVECT_1:def 3;
then a _|_ by RLVECT_1:5;
hence contradiction by A1, RLVECT_1:4; :: thesis: verum