reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;
reserve n,i,j for Nat;
reserve n for Nat;

theorem
  for A,B being Matrix of n,REAL st A is invertible & B is invertible
  holds A*B is invertible & Inv(A*B)=Inv(B)*(Inv A)
proof
  let A,B be Matrix of n,REAL;
  assume that
A1: A is invertible and
A2: B is invertible;
A3: (Inv(B)*(Inv(A)))*(A*B)=(Inv(B)*(Inv(A)))*A*B by Th28
    .=(Inv(B))*((Inv(A))*A)*B by Th28
    .=(Inv(B))*(1_Rmatrix(n))*B by A1,Def6
    .=(Inv(B))*B by Th71
    .=1_Rmatrix(n) by A2,Def6;
A4: (A*B)*(Inv(B)*(Inv(A)))=A*(B*(Inv(B)*(Inv(A)))) by Th28
    .=A*((B*(Inv(B)))*(Inv(A))) by Th28
    .=A*((1_Rmatrix(n))*(Inv(A))) by A2,Def6
    .=A*(1_Rmatrix(n))*(Inv(A)) by Th28
    .=A*(Inv(A)) by Th71
    .=1_Rmatrix(n) by A1,Def6;
  hence A*B is invertible by A3;
  hence thesis by A3,A4,Def6;
end;
