let a, b be Integer; :: thesis: for V being Z_Module
for v being VECTOR of V st V is Mult-cancelable & v <> 0. V & a * v = b * v holds
a = b
let V be Z_Module; :: thesis: for v being VECTOR of V st V is Mult-cancelable & v <> 0. V & a * v = b * v holds
a = b
let v be VECTOR of V; :: thesis: ( V is Mult-cancelable & v <> 0. V & a * v = b * v implies a = b )
assume A1: V is Mult-cancelable ; :: thesis: ( not v <> 0. V or not a * v = b * v or a = b )
assume that
A2: v <> 0. V and
A3: a * v = b * v ; :: thesis: a = b
0. V = (a * v) - (b * v) by A3, RLVECT_1:15
.= (a - b) * v by Th9 ;
then (- b) + a = 0 by A2, A1;
hence a = b ; :: thesis: verum