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 Th89:
  for x,y being FinSequence of REAL,A be Matrix of n,REAL st A is
  invertible & len x=n & len y=n holds x*A=y iff x=y*Inv(A)
proof
  let x,y be FinSequence of REAL, A be Matrix of n,REAL;
  assume that
A1: A is invertible and
A2: len x=n and
A3: len y=n;
A4: len A=n & len Inv A=n by MATRIX_0:24;
A5: width A=n by MATRIX_0:24;
  thus x*A=y implies x=y*Inv(A)
  proof
    assume
A6: x*A=y;
    thus x=x*(1_Rmatrix n) by A2,Th88
      .=x*(A*(Inv A)) by A1,Def6
      .=y*(Inv A) by A2,A5,A4,A6,Th57;
  end;
  assume
A7: x=y*(Inv A);
A8: width Inv A=n by MATRIX_0:24;
  thus y=y*(1_Rmatrix(n)) by A3,Th88
    .= y*((Inv A)*A) by A1,Def6
    .= x*A by A3,A4,A8,A7,Th57;
end;
