let a, b be real number ; :: thesis: for V being RealLinearSpace
for v being VECTOR of V st v <> 0. V & a * v = b * v holds
a = b
let V be RealLinearSpace; :: thesis: for v being VECTOR of V st v <> 0. V & a * v = b * v holds
a = b
let v be VECTOR of V; :: thesis: ( v <> 0. V & a * v = b * v implies a = b )
assume that
A1: v <> 0. V and
A2: a * v = b * v ; :: thesis: a = b
0. V = (a * v) - (b * v) by A2, Def13
.= (a - b) * v by Th49 ;
then (- b) + a = 0 by A1, Th24;
hence a = b ; :: thesis: verum