reconsider o = ~ the rmult of W as Function of [: the carrier of (opp K), the carrier of W:], the carrier of W ;

let M1, M2 be strict ModuleStr over opp K; :: thesis: ( ( for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M1 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) ) & ( for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M2 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) ) implies M1 = M2 )

assume that

A1: for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M1 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) and

A2: for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M2 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) ; :: thesis: M1 = M2

thus M1 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) by A1

.= M2 by A2 ; :: thesis: verum

let M1, M2 be strict ModuleStr over opp K; :: thesis: ( ( for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M1 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) ) & ( for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M2 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) ) implies M1 = M2 )

assume that

A1: for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M1 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) and

A2: for o being Function of [: the carrier of (opp K), the carrier of W:], the carrier of W st o = ~ the rmult of W holds

M2 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) ; :: thesis: M1 = M2

thus M1 = ModuleStr(# the carrier of W, the addF of W,(0. W),o #) by A1

.= M2 by A2 ; :: thesis: verum