let a, b be Real; :: thesis: for V being RealLinearSpace
for v being VECTOR of V st a <> b holds
{(a * v),(b * v)} is linearly-dependent
let V be RealLinearSpace; :: thesis: for v being VECTOR of V st a <> b holds
{(a * v),(b * v)} is linearly-dependent
let v be VECTOR of V; :: thesis: ( a <> b implies {(a * v),(b * v)} is linearly-dependent )
assume A1: a <> b ; :: thesis: {(a * v),(b * v)} is linearly-dependent
now :: thesis: {(a * v),(b * v)} is linearly-dependent
per cases ( v = 0. V or v <> 0. V ) ;
suppose A2: v <> 0. V ; :: thesis: {(a * v),(b * v)} is linearly-dependent
A3: (b * (a * v)) + ((- a) * (b * v)) = ((a * b) * v) + ((- a) * (b * v)) by RLVECT_1:def 7
.= ((a * b) * v) - (a * (b * v)) by Th3
.= ((a * b) * v) - ((a * b) * v) by RLVECT_1:def 7
.= 0. V by RLVECT_1:15 ;
A4: ( not b = 0 or not - a = 0 ) by A1;
a * v <> b * v by A1, A2, RLVECT_1:37;
hence {(a * v),(b * v)} is linearly-dependent by A3, A4, RLVECT_3:13; :: thesis: verum
end;
end;
end;
hence {(a * v),(b * v)} is linearly-dependent ; :: thesis: verum