reserve i,j for Nat;

theorem
  for K being Ring,n,m,k being Nat, M1 being Matrix of n,m,K
, M2 being Matrix of m,k,K st width M1=len M2 & 0< len M1 & 0<len M2 holds M1*
  M2 is Matrix of n,k,K
proof
  let K be Ring,n,m,k be Nat, M1 be Matrix of n,m,K, M2 be Matrix
  of m,k,K;
  assume that
A1: width M1=len M2 and
A2: 0< len M1 and
A3: 0< len M2;
  width M1=m by A1,MATRIX_0:def 2;
  then
A4: len M1=n & width M2=k by A1,A3,MATRIX_0:20,def 2;
  len (M1*M2)=len M1 & width (M1*M2)=width M2 by A1,MATRIX_3:def 4;
  hence thesis by A2,A4,MATRIX_0:20;
end;
