let M1, M2 be Matrix of REAL; :: thesis: ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds

M1 * (i,j) = |.(M * (i,j)).| ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds

M2 * (i,j) = |.(M * (i,j)).| ) implies M1 = M2 )

assume that

A6: len M1 = len M and

A7: width M1 = width M and

A8: for i, j being Nat st [i,j] in Indices M holds

M1 * (i,j) = |.(M * (i,j)).| and

A9: ( len M2 = len M & width M2 = width M ) and

A10: for i, j being Nat st [i,j] in Indices M holds

M2 * (i,j) = |.(M * (i,j)).| ; :: thesis: M1 = M2

M1 * (i,j) = |.(M * (i,j)).| ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds

M2 * (i,j) = |.(M * (i,j)).| ) implies M1 = M2 )

assume that

A6: len M1 = len M and

A7: width M1 = width M and

A8: for i, j being Nat st [i,j] in Indices M holds

M1 * (i,j) = |.(M * (i,j)).| and

A9: ( len M2 = len M & width M2 = width M ) and

A10: for i, j being Nat st [i,j] in Indices M holds

M2 * (i,j) = |.(M * (i,j)).| ; :: thesis: M1 = M2

now :: thesis: for i, j being Nat st [i,j] in Indices M1 holds

M1 * (i,j) = M2 * (i,j)

hence
M1 = M2
by A6, A7, A9, MATRIX_0:21; :: thesis: verumM1 * (i,j) = M2 * (i,j)

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: dom M1 = dom M by A6, FINSEQ_3:29;

hence M1 * (i,j) = |.(M * (i,j)).| by A7, A8, A11

.= M2 * (i,j) by A7, A10, A11, A12 ;

:: thesis: verum

end;assume A11: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)

A12: dom M1 = dom M by A6, FINSEQ_3:29;

hence M1 * (i,j) = |.(M * (i,j)).| by A7, A8, A11

.= M2 * (i,j) by A7, A10, A11, A12 ;

:: thesis: verum