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
A3: len M1 = (2 |^ n) + 3 and
A4: len M1 = width M1 and
A5: for i, j being Nat st [i,j] in Indices M1 holds
M1 * i,j = H₁(i,j) and
A6: len M2 = (2 |^ n) + 3 and
A7: len M2 = width M2 and
A8: for i, j being Nat st [i,j] in Indices M2 holds
M2 * i,j = H₁(i,j) ; :: thesis: M1 = M2

now

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * i,j = M2 * i,j )
assume A9: [i,j] in Indices M1 ; :: thesis: M1 * i,j = M2 * i,j
A10: M1 is Matrix of (2 |^ n) + 3,(2 |^ n) + 3,the carrier of (TOP-REAL 2) by A1, A3, A4, MATRIX_1:20;
M2 is Matrix of (2 |^ n) + 3,(2 |^ n) + 3,the carrier of (TOP-REAL 2) by A1, A6, A7, MATRIX_1:20;
then A11: [i,j] in Indices M2 by A9, A10, MATRIX_1:27;
thus M1 * i,j = H₁(i,j) by A5, A9
.= M2 * i,j by A8, A11 ; :: thesis: verum

end;

hence M1 = M2 by A3, A4, A6, A7, MATRIX_1:21; :: thesis: verum