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 Th43:
  f,g equal_at m,n & g,h equal_at m,n implies f,h equal_at m,n
proof
  assume that
A1: f,g equal_at m,n and
A2: g,h equal_at m,n;
A3: dom f = dom g by A1;
A4: now
    let k;
    assume
A5: k in dom f & m <=k & k <= n;
    hence f.k=g.k by A1
      .=h.k by A2,A3,A5;
  end;
  dom g = dom h by A2;
  hence thesis by A3,A4;
end;
