deffunc H₁( Nat, Nat) -> Element of the carrier of K = a * (M * $1,$2);
consider N being Matrix of len M, width M,K such that
A1: for i, j being Nat st [i,j] in Indices N holds
N * i,j = H₁(i,j) from MATRIX_1:sch 1();
take N ; :: thesis: ( len N = len M & width N = width M & ( for i, j being Nat st [i,j] in Indices M holds
N * i,j = a * (M * i,j) ) )
A2: len N = len M by MATRIX_1:def 3;

A3: now

per cases ( len M = 0 or len M > 0 ) ;

case A4: len M = 0 ; :: thesis: width N = width M

then width N = 0 by A2, MATRIX_1:def 4;
hence width N = width M by A4, MATRIX_1:def 4; :: thesis: verum

end;

case len M > 0 ; :: thesis: width N = width M

hence width N = width M by MATRIX_1:24; :: thesis: verum

end;

end;

end;

dom M = dom N by A2, FINSEQ_3:31;
then ( Indices N = [:(dom M),(Seg (width M)):] & Indices M = [:(dom M),(Seg (width M)):] ) by A3, MATRIX_1:def 5;
hence ( len N = len M & width N = width M & ( for i, j being Nat st [i,j] in Indices M holds
N * i,j = a * (M * i,j) ) ) by A1, A3, MATRIX_1:def 3; :: thesis: verum