let n be Nat; :: thesis: for K being Field

for f being FinSequence of K

for M being Matrix of n,K

for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let K be Field; :: thesis: for f being FinSequence of K

for M being Matrix of n,K

for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let f be FinSequence of K; :: thesis: for M being Matrix of n,K

for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let M be Matrix of n,K; :: thesis: for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let p be Element of Permutations n; :: thesis: for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let i be Nat; :: thesis: ( len f = n & i in Seg n implies mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i) )

assume that

A1: len f = n and

A2: i in Seg n ; :: thesis: mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

reconsider N = n as Element of NAT by ORDINAL1:def 12;

set KK = the carrier of K;

set C = Matrix_of_Cofactor M;

reconsider Tp = f, TL = Line ((Matrix_of_Cofactor M),i) as Element of N -tuples_on the carrier of K by A1, FINSEQ_2:92, MATRIX_0:24;

set R = RLine (M,i,f);

set LL = LaplaceExpL ((RLine (M,i,f)),i);

set MLT = mlt (TL,Tp);

A3: len (LaplaceExpL ((RLine (M,i,f)),i)) = n by Def7;

hence mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i) by A3, A4; :: thesis: verum

for f being FinSequence of K

for M being Matrix of n,K

for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let K be Field; :: thesis: for f being FinSequence of K

for M being Matrix of n,K

for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let f be FinSequence of K; :: thesis: for M being Matrix of n,K

for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let M be Matrix of n,K; :: thesis: for p being Element of Permutations n

for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let p be Element of Permutations n; :: thesis: for i being Nat st len f = n & i in Seg n holds

mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

let i be Nat; :: thesis: ( len f = n & i in Seg n implies mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i) )

assume that

A1: len f = n and

A2: i in Seg n ; :: thesis: mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i)

reconsider N = n as Element of NAT by ORDINAL1:def 12;

set KK = the carrier of K;

set C = Matrix_of_Cofactor M;

reconsider Tp = f, TL = Line ((Matrix_of_Cofactor M),i) as Element of N -tuples_on the carrier of K by A1, FINSEQ_2:92, MATRIX_0:24;

set R = RLine (M,i,f);

set LL = LaplaceExpL ((RLine (M,i,f)),i);

set MLT = mlt (TL,Tp);

A3: len (LaplaceExpL ((RLine (M,i,f)),i)) = n by Def7;

A4: now :: thesis: for j being Nat st 1 <= j & j <= n holds

(mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j

len (mlt (TL,Tp)) = n
by CARD_1:def 7;(mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j

A5:
dom (LaplaceExpL ((RLine (M,i,f)),i)) = Seg n
by A3, FINSEQ_1:def 3;

A6: n = width M by MATRIX_0:24;

let j be Nat; :: thesis: ( 1 <= j & j <= n implies (mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j )

assume that

A7: 1 <= j and

A8: j <= n ; :: thesis: (mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j

A9: j in Seg n by A7, A8;

n = width (Matrix_of_Cofactor M) by MATRIX_0:24;

then A10: (Line ((Matrix_of_Cofactor M),i)) . j = (Matrix_of_Cofactor M) * (i,j) by A9, MATRIX_0:def 7;

Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;

then [i,j] in Indices M by A2, A9, ZFMISC_1:87;

then A11: (RLine (M,i,f)) * (i,j) = f . j by A1, A6, MATRIX11:def 3;

Indices (Matrix_of_Cofactor M) = [:(Seg n),(Seg n):] by MATRIX_0:24;

then [i,j] in Indices (Matrix_of_Cofactor M) by A2, A9, ZFMISC_1:87;

then (Line ((Matrix_of_Cofactor M),i)) . j = Cofactor (M,i,j) by A10, Def6;

then A12: (mlt (TL,Tp)) . j = (Cofactor (M,i,j)) * ((RLine (M,i,f)) * (i,j)) by A9, A11, FVSUM_1:61;

Cofactor (M,i,j) = Cofactor ((RLine (M,i,f)),i,j) by A2, A9, Th15;

hence (mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j by A9, A5, A12, Def7; :: thesis: verum

end;A6: n = width M by MATRIX_0:24;

let j be Nat; :: thesis: ( 1 <= j & j <= n implies (mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j )

assume that

A7: 1 <= j and

A8: j <= n ; :: thesis: (mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j

A9: j in Seg n by A7, A8;

n = width (Matrix_of_Cofactor M) by MATRIX_0:24;

then A10: (Line ((Matrix_of_Cofactor M),i)) . j = (Matrix_of_Cofactor M) * (i,j) by A9, MATRIX_0:def 7;

Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;

then [i,j] in Indices M by A2, A9, ZFMISC_1:87;

then A11: (RLine (M,i,f)) * (i,j) = f . j by A1, A6, MATRIX11:def 3;

Indices (Matrix_of_Cofactor M) = [:(Seg n),(Seg n):] by MATRIX_0:24;

then [i,j] in Indices (Matrix_of_Cofactor M) by A2, A9, ZFMISC_1:87;

then (Line ((Matrix_of_Cofactor M),i)) . j = Cofactor (M,i,j) by A10, Def6;

then A12: (mlt (TL,Tp)) . j = (Cofactor (M,i,j)) * ((RLine (M,i,f)) * (i,j)) by A9, A11, FVSUM_1:61;

Cofactor (M,i,j) = Cofactor ((RLine (M,i,f)),i,j) by A2, A9, Th15;

hence (mlt (TL,Tp)) . j = (LaplaceExpL ((RLine (M,i,f)),i)) . j by A9, A5, A12, Def7; :: thesis: verum

hence mlt ((Line ((Matrix_of_Cofactor M),i)),f) = LaplaceExpL ((RLine (M,i,f)),i) by A3, A4; :: thesis: verum