let V be RealLinearSpace; :: thesis: for p, q, y being Element of V
for a being Real st not are_Prop p,q & y = a * q & a <> 0 holds
not are_Prop p,y
let p, q, y be Element of V; :: thesis: for a being Real st not are_Prop p,q & y = a * q & a <> 0 holds
not are_Prop p,y
let a be Real; :: thesis: ( not are_Prop p,q & y = a * q & a <> 0 implies not are_Prop p,y )
assume A1: ( not are_Prop p,q & y = a * q & a <> 0 ) ; :: thesis: not are_Prop p,y
assume are_Prop p,y ; :: thesis: contradiction
then consider b being Real such that
A2: ( b <> 0 & p = b * y ) by ANPROJ_1:5;
A3: p = (b * a) * q by A1, A2, RLVECT_1:def 9;
b * a <> 0 by A1, A2, XCMPLX_1:6;
hence contradiction by A1, A3, ANPROJ_1:5; :: thesis: verum