let M1, M2 be Matrix of (TOP-REAL 2); :: thesis: ( len M1 = (2 |^ n) + 3 & len M1 = width M1 & ( for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (j - 2)))]| ) & len M2 = (2 |^ n) + 3 & len M2 = width M2 & ( for i, j being Nat st [i,j] in Indices M2 holds
M2 * (i,j) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (j - 2)))]| ) implies M1 = M2 )
assume that
A2: len M1 = (2 |^ n) + 3 and
A3: len M1 = width M1 and
A4: for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = H₁(i,j) and
A5: len M2 = (2 |^ n) + 3 and
A6: len M2 = width M2 and
A7: for i, j being Nat st [i,j] in Indices M2 holds
M2 * (i,j) = H₁(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)

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume A8: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
A9: M1 is Matrix of (2 |^ n) + 3,(2 |^ n) + 3, the carrier of (TOP-REAL 2) by A2, A3, MATRIX_0:20;
M2 is Matrix of (2 |^ n) + 3,(2 |^ n) + 3, the carrier of (TOP-REAL 2) by A5, A6, MATRIX_0:20;
then A10: [i,j] in Indices M2 by A8, A9, MATRIX_0:26;
thus M1 * (i,j) = H₁(i,j) by A4, A8
.= M2 * (i,j) by A7, A10 ; :: thesis: verum

end;

hence M1 = M2 by A2, A3, A5, A6, MATRIX_0:21; :: thesis: verum