let I be non empty set ; for F being Group-like associative multMagma-Family of I
for i being Element of I
for g1 being Element of (product F)
for z1 being Element of (F . i) st g1 = (1_ (product F)) +* (i,z1) holds
g1 " = (1_ (product F)) +* (i,(z1 "))
let F be Group-like associative multMagma-Family of I; for i being Element of I
for g1 being Element of (product F)
for z1 being Element of (F . i) st g1 = (1_ (product F)) +* (i,z1) holds
g1 " = (1_ (product F)) +* (i,(z1 "))
let i be Element of I; for g1 being Element of (product F)
for z1 being Element of (F . i) st g1 = (1_ (product F)) +* (i,z1) holds
g1 " = (1_ (product F)) +* (i,(z1 "))
let g1 be Element of (product F); for z1 being Element of (F . i) st g1 = (1_ (product F)) +* (i,z1) holds
g1 " = (1_ (product F)) +* (i,(z1 "))
let z1 be Element of (F . i); ( g1 = (1_ (product F)) +* (i,z1) implies g1 " = (1_ (product F)) +* (i,(z1 ")) )
assume A1:
g1 = (1_ (product F)) +* (i,z1)
; g1 " = (1_ (product F)) +* (i,(z1 "))
set x1 = g1;
A2:
( g1 = g1 & dom g1 = I & g1 . i = z1 & ( for j being Element of I st j <> i holds
g1 . j = 1_ (F . j) ) )
by Th1, A1;
set x12 = g1 " ;
the carrier of (product F) = product (Carrier F)
by GROUP_7:def 2;
then A3:
dom (g1 ") = I
by PARTFUN1:def 2;
A4:
(g1 ") . i = z1 "
by A2, GROUP_7:8;
A5:
for j being Element of I st i <> j holds
(g1 ") . j = 1_ (F . j)
thus
g1 " = (1_ (product F)) +* (i,(z1 "))
by A3, A4, A5, Th1; verum