let I be non empty set ; :: thesis: 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; :: thesis: 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; :: thesis: 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); :: thesis: 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); :: thesis: ( g1 = (1_ (product F)) +* (i,z1) implies g1 " = (1_ (product F)) +* (i,(z1 ")) )

assume A1: g1 = (1_ (product F)) +* (i,z1) ; :: thesis: 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)

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; :: thesis: 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; :: thesis: 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); :: thesis: 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); :: thesis: ( g1 = (1_ (product F)) +* (i,z1) implies g1 " = (1_ (product F)) +* (i,(z1 ")) )

assume A1: g1 = (1_ (product F)) +* (i,z1) ; :: thesis: 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)

proof

thus
g1 " = (1_ (product F)) +* (i,(z1 "))
by A3, A4, A5, Th1; :: thesis: verum
let j be Element of I; :: thesis: ( i <> j implies (g1 ") . j = 1_ (F . j) )

assume i <> j ; :: thesis: (g1 ") . j = 1_ (F . j)

then g1 . j = 1_ (F . j) by Th1, A1;

hence (g1 ") . j = (1_ (F . j)) " by GROUP_7:8

.= 1_ (F . j) by GROUP_1:8 ;

:: thesis: verum

end;assume i <> j ; :: thesis: (g1 ") . j = 1_ (F . j)

then g1 . j = 1_ (F . j) by Th1, A1;

hence (g1 ") . j = (1_ (F . j)) " by GROUP_7:8

.= 1_ (F . j) by GROUP_1:8 ;

:: thesis: verum