reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;
reserve p,q for FinSequence;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G;

theorem Th9:
  p^q is oriented Chain of G implies p is oriented Chain of G & q
  is oriented Chain of G
proof
  assume
A1: p^q is oriented Chain of G;
  set r=p^q;
A2: len r = len p + len q by FINSEQ_1:22;
A3: now
    let n;
    assume that
A4: 1 <= n and
A5: n < len q;
    set m=len p + n;
    n <= m by NAT_1:11;
    then
A6: 1 <= m by A4,XXREAL_0:2;
    n+1 <= len q by A5,NAT_1:13;
    then
A7: q.(n+1)=r.(len p+(n+1)) by Lm2,NAT_1:11
      .=r.(m+1);
    m < len r by A2,A5,XREAL_1:8;
    then (the Source of G).(r.(m+1))=(the Target of G).(r.m) by A1,A6,
GRAPH_1:def 15;
    hence (the Source of G).(q.(n+1))=(the Target of G).(q.n) by A4,A5,A7,Lm2;
  end;
  now
    let n;
    assume that
A8: 1 <= n and
A9: n < len p;
    n+1 <= len p by A9,NAT_1:13;
    then
A10: p.(n+1)=r.(n+1) by Lm1,NAT_1:11;
    len p <= len r by A2,NAT_1:11;
    then n < len r by A9,XXREAL_0:2;
    then (the Source of G).(r.(n+1))=(the Target of G).(r.n) by A1,A8,
GRAPH_1:def 15;
    hence (the Source of G).(p.(n+1))=(the Target of G).(p.n) by A8,A9,A10,Lm1;
  end;
  hence thesis by A1,A3,Th8,GRAPH_1:def 15;
end;
