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;

theorem Th17:
  i <> k & i in dom f implies ((f,i):=(k,r)).i = k
proof
  assume that
A1: i <> k and
A2: i in dom f;
  set fik = (f,i):=k;
  thus ((f,i):=(k,r)).i =fik.i by A1,FUNCT_7:32
    .=k by A2,FUNCT_7:31;
end;
