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_0: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_0:def 2;
A3: now :: thesis: ( ( len M = 0 & width N = width M ) or ( len M > 0 & width N = width M ) )
per cases ( len M = 0 or len M > 0 ) ;
end;
end;
dom M = dom N by A2, FINSEQ_3:29;
then Indices N = [:(dom M),(Seg (width M)):] by A3;
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_0:def 2; :: thesis: verum