reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G,
  p,q for oriented Chain of G,
  W for Function,
  U,V,e,ee for set,
  v1,v2,v3,v4 for Vertex of G;
reserve G for finite Graph,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Vertex of G;
reserve G for finite oriented Graph,
  P,Q for oriented Chain of G,
  W for Function of (the carrier' of G), Real>=0,
  v1,v2,v3,v4 for Vertex of G;
reserve f,g,h for Element of REAL*,
  r for Real;
reserve G for oriented Graph,
  v1,v2 for Vertex of G,
  W for Function of (the carrier' of G), Real>=0;

theorem Th44:
  (repeat(Relax(n)*findmin(n))).i.f, (repeat(Relax(n)*findmin(n)))
  .(i+1).f equal_at 3*n+1,n*n+3*n
proof
  set R=Relax(n), M=findmin(n), ff=(repeat (R*M)).i.f;
  set Fi1=(repeat (R*M)).(i+1).f;
A1: now
    let k;
    assume that
A2: k in dom ff and
A3: 3*n+1 <= k and
A4: k <= n*n+3*n;
A5: k > 3*n by A3,NAT_1:13;
A6: k in dom (M.ff) by A2,Th33;
A7: k < n*n+3*n+1 by A4,NAT_1:13;
    3*n >= n by Lm6;
    then
A8: k > n by A5,XXREAL_0:2;
    thus Fi1.k=(R.(M.ff)).k by Th22
      .=(M.ff).k by A5,A6,Th36
      .=ff.k by A7,A8,Th31;
  end;
  dom (Fi1) = dom ff by Th37;
  hence thesis by A1;
end;
