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 Th96:
  for A being Matrix of n,REAL st n>0 & (for y being FinSequence
  of REAL st len y=n holds (ex x being FinSequence of REAL st len x=n & A*x=y))
  holds ex B being Matrix of n,REAL st A*B=1_Rmatrix(n)
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 x being
  FinSequence of REAL st len x=n & A*x=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 consider x0 being FinSequence of REAL such that
A3: len x0=n and
A4: A*x0=y by A2;
    x0*A@=A*x0 by A1,A3,Th95;
    hence thesis by A3,A4;
  end;
  then consider B being Matrix of n,REAL such that
A5: B*(A@)=1_Rmatrix n by Th93;
  (B*(A@))@=1_Rmatrix n by A5,Th64;
  then (A@@)*B@=1_Rmatrix n by Th30;
  then A*(B@)=1_Rmatrix n by Th29;
  hence thesis;
end;
