deffunc H₁( Nat, Nat) -> Element of REAL = In (|.(M * ($1,$2)).|,REAL);
consider M1 being Matrix of len M, width M,REAL such that
A1: for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = H₁(i,j) from MATRIX_0:sch 1();
take M1 ; :: thesis: ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = |.(M * (i,j)).| ) )
A2: len M1 = len M by MATRIX_0:def 2;

A3: now :: thesis: ( ( len M = 0 & width M1 = width M ) or ( len M > 0 & width M1 = width M ) )

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

case A4: len M = 0 ; :: thesis: width M1 = width M

then width M1 = 0 by A2, MATRIX_0:def 3;
hence width M1 = width M by A4, MATRIX_0:def 3; :: thesis: verum

end;

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

hence width M1 = width M by MATRIX_0:23; :: thesis: verum

end;

end;

end;

thus ( len M1 = len M & width M1 = width M ) by A3, A2; :: thesis: for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = |.(M * (i,j)).|
let i, j be Nat; :: thesis: ( [i,j] in Indices M implies M1 * (i,j) = |.(M * (i,j)).| )
assume A5: [i,j] in Indices M ; :: thesis: M1 * (i,j) = |.(M * (i,j)).|
dom M = dom M1 by A2, FINSEQ_3:29;
hence M1 * (i,j) = In (|.(M * (i,j)).|,REAL) by A1, A3, A5
.= |.(M * (i,j)).| ;
:: thesis: verum