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 that

A1: not are_Prop p,q and

A2: ( 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

A3: ( b <> 0 & p = b * y ) by ANPROJ_1:1;

( p = (b * a) * q & b * a <> 0 ) by A2, A3, RLVECT_1:def 7, XCMPLX_1:6;

hence contradiction by A1, ANPROJ_1:1; :: thesis: verum

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 that

A1: not are_Prop p,q and

A2: ( 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

A3: ( b <> 0 & p = b * y ) by ANPROJ_1:1;

( p = (b * a) * q & b * a <> 0 ) by A2, A3, RLVECT_1:def 7, XCMPLX_1:6;

hence contradiction by A1, ANPROJ_1:1; :: thesis: verum