deffunc H₁( Nat, Nat) -> Element of COMPLEX = ((x . $1) * (M * ($1,$2))) * ((y . $2) *');
consider M1 being Matrix of len M, width M, COMPLEX 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 2;

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 3;
hence width M1 = len y by A2, A6, MATRIX_1:def 3; :: thesis: verum

end;

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

hence width M1 = len y by A2, MATRIX_1:23; :: thesis: verum

end;

end;

end;

dom M = dom M1 by A1, A4, FINSEQ_3:29;
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, A2, A3, A5, MATRIX_1:def 2; :: thesis: verum