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
A7: ( len M1 = n & width M1 = n ) and
A8: 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
A9: ( len M2 = n & width M2 = n ) and
A10: 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

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * i,j = M2 * i,j )
assume A11: [i,j] in Indices M1 ; :: thesis: M1 * i,j = M2 * i,j
A12: Indices M1 = [:(Seg n),(Seg n):] by A7, FINSEQ_1:def 3
.= Indices M2 by A9, 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 A8, A10, A11, A12;
hence M1 * i,j = M2 * i,j ; :: thesis: verum

end;

hence M1 = M2 by A7, A9, MATRIX_1:21; :: thesis: verum