let p be prime Element of INT.Ring; :: thesis: for V being free Z_Module

for s being Element of V

for a being Element of INT.Ring

for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let V be free Z_Module; :: thesis: for s being Element of V

for a being Element of INT.Ring

for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let s be Element of V; :: thesis: for a being Element of INT.Ring

for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let a be Element of INT.Ring; :: thesis: for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let b be Element of (GF p); :: thesis: ( a = b implies b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s)) )

set t = ZMtoMQV (V,p,s);

assume A1: a = b ; :: thesis: b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

A2: ZMtoMQV (V,p,s) = s + (p (*) V) ;

reconsider t1 = ZMtoMQV (V,p,s) as Element of (VectQuot (V,(p (*) V))) ;

A3: s + (p (*) V) is Element of CosetSet (V,(p (*) V)) by A2, VECTSP10:def 6;

reconsider i = b as Nat ;

A4: b = i mod p by NAT_1:44, NAT_D:24;

i in INT by INT_1:def 2;

then reconsider i = i as Element of INT.Ring ;

thus b * (ZMtoMQV (V,p,s)) = (i mod p) * t1 by A4, ZMODUL02:def 11

.= (lmultCoset (V,(p (*) V))) . ((i mod p),(s + (p (*) V))) by VECTSP10:def 6

.= ZMtoMQV (V,p,(a * s)) by A1, A4, A3, VECTSP10:def 5 ; :: thesis: verum

for s being Element of V

for a being Element of INT.Ring

for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let V be free Z_Module; :: thesis: for s being Element of V

for a being Element of INT.Ring

for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let s be Element of V; :: thesis: for a being Element of INT.Ring

for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let a be Element of INT.Ring; :: thesis: for b being Element of (GF p) st a = b holds

b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

let b be Element of (GF p); :: thesis: ( a = b implies b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s)) )

set t = ZMtoMQV (V,p,s);

assume A1: a = b ; :: thesis: b * (ZMtoMQV (V,p,s)) = ZMtoMQV (V,p,(a * s))

A2: ZMtoMQV (V,p,s) = s + (p (*) V) ;

reconsider t1 = ZMtoMQV (V,p,s) as Element of (VectQuot (V,(p (*) V))) ;

A3: s + (p (*) V) is Element of CosetSet (V,(p (*) V)) by A2, VECTSP10:def 6;

reconsider i = b as Nat ;

A4: b = i mod p by NAT_1:44, NAT_D:24;

i in INT by INT_1:def 2;

then reconsider i = i as Element of INT.Ring ;

thus b * (ZMtoMQV (V,p,s)) = (i mod p) * t1 by A4, ZMODUL02:def 11

.= (lmultCoset (V,(p (*) V))) . ((i mod p),(s + (p (*) V))) by VECTSP10:def 6

.= ZMtoMQV (V,p,(a * s)) by A1, A4, A3, VECTSP10:def 5 ; :: thesis: verum