deffunc H₁( Nat, Nat) -> Element of REAL = ((x . $1) * (M * ($1,$2))) * (y . $2);
consider M1 being Matrix of len M, width M, REAL such that
A3: for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = H₁(i,j) from MATRIX_1:sch 1();
take M1 ; :: thesis: ( len M1 = len x & width M1 = len y & ( for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * (y . j) ) )
A4: len M1 = len x by A1, MATRIX_1:def 3;

A5: now

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

case A6: len M = 0 ; :: thesis: width M1 = len y

then width M1 = 0 by A1, A4, MATRIX_1:def 4;
hence width M1 = len y by A2, A6, MATRIX_1:def 4; :: thesis: verum

end;

case len M > 0 ; :: thesis: width M1 = len y

hence width M1 = len y by A2, MATRIX_1:24; :: thesis: verum

end;

end;

end;

A7: Indices M = [:(dom M),(Seg (width M)):] by MATRIX_1:def 5;
dom M = dom M1 by A1, A4, FINSEQ_3:31;
then Indices M1 = [:(dom M),(Seg (width M)):] by A2, A5, MATRIX_1:def 5;
hence ( len M1 = len x & width M1 = len y & ( for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * (y . j) ) ) by A1, A3, A5, A7, MATRIX_1:def 3; :: thesis: verum