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 Th87:
  for n being Nat,x,y being FinSequence of REAL, A be
Matrix of n,REAL st A is invertible & len x=n & len y=n & n>0 holds A*x=y iff x
  =Inv(A)*y
proof
  let n be Nat,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 and
A4: n>0;
A5: width A=n & width (Inv(A))=n by MATRIX_0:24;
A6: len A=n by MATRIX_0:24;
  thus A*x=y implies x=Inv(A)*y
  proof
    assume
A7: A*x=y;
    thus x=(1_Rmatrix(n))*x by A2,A4,Th86
      .=((Inv A)*A) *x by A1,Def6
      .=Inv(A)*y by A2,A4,A6,A5,A7,Th59;
  end;
A8: len (Inv(A))=n by MATRIX_0:24;
  x=Inv(A)*y implies A*x=y
  proof
    assume
A9: x=Inv(A)*y;
    thus y=(1_Rmatrix(n))*y by A3,A4,Th86
      .=(A*(Inv A))*y by A1,Def6
      .=A*x by A3,A4,A8,A5,A9,Th59;
  end;
  hence thesis;
end;
