let a be real number ; :: thesis: for V being RealLinearSpace
for v, w being VECTOR of V st a <> 0 & a * v = a * w holds
v = w
let V be RealLinearSpace; :: thesis: for v, w being VECTOR of V st a <> 0 & a * v = a * w holds
v = w
let v, w be VECTOR of V; :: thesis: ( a <> 0 & a * v = a * w implies v = w )
assume that
A1: a <> 0 and
A2: a * v = a * w ; :: thesis: v = w
0. V = (a * v) - (a * w) by A2, Def13
.= a * (v - w) by Th48 ;
then v - w = 0. V by A1, Th24;
hence v = w by Th35; :: thesis: verum