let M1, M2 be Matrix of K; :: thesis: ( len M1 = n & width M1 = n & ( for i, j being Nat st [i,j] in Indices M1 holds
( ( i = j implies M1 * (i,j) = L ) & ( i + 1 = j implies M1 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M1 * (i,j) = 0. K ) ) ) & len M2 = n & width M2 = n & ( for i, j being Nat st [i,j] in Indices M2 holds
( ( i = j implies M2 * (i,j) = L ) & ( i + 1 = j implies M2 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M2 * (i,j) = 0. K ) ) ) implies M1 = M2 )
assume that
A6: ( len M1 = n & width M1 = n ) and
A7: for i, j being Nat st [i,j] in Indices M1 holds
( ( i = j implies M1 * (i,j) = L ) & ( i + 1 = j implies M1 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M1 * (i,j) = 0. K ) ) and
A8: ( len M2 = n & width M2 = n ) and
A9: for i, j being Nat st [i,j] in Indices M2 holds
( ( i = j implies M2 * (i,j) = L ) & ( i + 1 = j implies M2 * (i,j) = 1_ K ) & ( i <> j & i + 1 <> j implies M2 * (i,j) = 0. K ) ) ; :: thesis: M1 = M2

now :: thesis: for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j)

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume A10: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
A11: Indices M1 = [:(Seg n),(Seg n):] by A6, FINSEQ_1:def 3
.= Indices M2 by A8, FINSEQ_1:def 3 ;
( i = j or i + 1 = j or ( i <> j & i + 1 <> j ) ) ;
then ( ( M1 * (i,j) = L & M2 * (i,j) = L ) or ( M1 * (i,j) = 1_ K & M2 * (i,j) = 1_ K ) or ( M1 * (i,j) = 0. K & M2 * (i,j) = 0. K ) ) by A7, A9, A10, A11;
hence M1 * (i,j) = M2 * (i,j) ; :: thesis: verum

end;

hence M1 = M2 by A6, A8, MATRIX_0:21; :: thesis: verum