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 being Matrix of n,REAL st n>0 & (for y being FinSequence of REAL
st len y=n holds (ex x1,x2 being FinSequence of REAL st len x1=n & len x2=n & A
  *x1=y & x2*A=y)) holds A is invertible
proof
  let A be Matrix of n,REAL;
  assume that
A1: n>0 and
A2: for y being FinSequence of REAL st len y=n holds ex x1,x2 being
  FinSequence of REAL st len x1=n & len x2=n & A*x1=y & x2*A=y;
  for y being FinSequence of REAL st len y=n holds ex x being FinSequence
  of REAL st len x=n & x*A=y
  proof
    let y be FinSequence of REAL;
    assume len y=n;
    then ex x1,x2 being FinSequence of REAL st len x1=n & len x2= n & A*x1=y &
    x2*A=y by A2;
    hence thesis;
  end;
  then
A3: ex B1 being Matrix of n,REAL st B1*A=1_Rmatrix(n) by Th93;
  for y being FinSequence of REAL st len y=n holds ex x being FinSequence
  of REAL st len x=n & A*x=y
  proof
    let y be FinSequence of REAL;
    assume len y=n;
    then ex x1,x2 being FinSequence of REAL st len x1=n & len x2= n & A*x1=y &
    x2*A=y by A2;
    hence thesis;
  end;
  then ex B2 being Matrix of n,REAL st A*B2=1_Rmatrix(n) by A1,Th96;
  hence thesis by A3,Th80;
end;
