 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;
reserve f,f1,f2 for n-element real-valued FinSequence,
        p,p1,p2 for Point of TOP-REAL n,
        M,M1,M2 for Matrix of n,m,F_Real,
        A,B for Matrix of n,F_Real;

theorem
  the_rank_of M = n implies
    ex L be Real st L > 0 & for f holds |.f.| <= L*|.(Mx2Tran M).f.|
proof
  assume that
   A1: the_rank_of M=n;
  per cases;
  suppose A2: n<>0;
   consider N be Matrix of m-' n,m,F_Real such that
    A3: the_rank_of(M^N)=m by A1,Th16;
   width M=m by A2,MATRIX13:1;
   then m-n>=0 by A1,MATRIX13:74,XREAL_1:48;
   then A4: m-n=m-' n by XREAL_0:def 2;
   then reconsider MN=M^N as Matrix of m,F_Real;
   A5: dom id TOP-REAL m=[#]TOP-REAL m;
   set mn=(m-' n) |->0;
   A6: m=0 iff m=0;
   sqr mn=(m-' n) |->((0 qua Real)^2) by RVSUM_1:56
    .=(m-' n) |->(0*0);
   then A7: Sum sqr mn=(m-' n)*(0 qua Nat) by RVSUM_1:80;
   set MN1=MN~;
   consider L be Real such that
    A8: L>0 and
    A9: for f be m-element real-valued FinSequence holds
     |.(Mx2Tran MN1).f.|<=L*|.f.| by Th45;
   take L;
   thus L>0 by A8;
   let f be n-element real-valued FinSequence;
   set fm=f^mn;
   set Mfm=(Mx2Tran MN).fm;
   A10: Mfm =(Mx2Tran M).f by A4,Th35;
   Det MN<>0.F_Real by A3,MATRIX13:83;
   then A11: MN is invertible by LAPLACE:34;
A12:  width MN = m by MATRIX_0:24;
   reconsider MN2=MN1 as Matrix of width MN,m,F_Real
            by A12;
   A13: width MN1=m & len MN=m by MATRIX_0:24;
   A14: (Mx2Tran MN1)*(Mx2Tran MN)=(Mx2Tran MN2)*(Mx2Tran MN) by MATRIX_0:24
    .=Mx2Tran(MN*MN2) by A6,Th32
    .=Mx2Tran 1.(F_Real,m) by A11,A13,MATRIX14:18
    .=id TOP-REAL m by Th33;
   sqr fm=sqr f^sqr mn by RVSUM_1:144;
   then Sum sqr fm=Sum sqr f+Sum sqr mn by RVSUM_1:75
    .=Sum sqr f by A7;
   then A15: |.fm.|  =|.f.|;
   A16: fm is Point of TOP-REAL m by A4,Lm2;
   then fm=(id TOP-REAL m).fm
    .=(Mx2Tran MN1).Mfm by A16,A14,A5,FUNCT_1:12;
   hence thesis by A9,A10,A15;
  end;
  suppose A17: n=0;
   A18: |. 0*n .| = 0 by EUCLID:7;
   take L=jj;
   thus thesis by A17,A18;
  end;
end;
