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 Th80:
  for A,B1,B2 being Matrix of n,REAL st B1*A=1_Rmatrix(n) & A*B2=
  1_Rmatrix(n) holds B1=B2 & A is invertible
proof
  let A,B1,B2 be Matrix of n,REAL;
  assume that
A1: B1*A=1_Rmatrix(n) and
A2: A*B2=1_Rmatrix(n);
A3: B1*A*B2=B1*(A*B2) by Th28
    .= B1 by A2,Th71;
  hence B1=B2 by A1,Th70;
  B1*A*B2=B2 by A1,Th70;
  hence thesis by A1,A2,A3;
end;
