reserve x,y,c for set;

theorem Th8:
  for x,y,c being non pair set holds x in InputVertices BorrowStr(x
,y,c) & y in InputVertices BorrowStr(x,y,c) & c in InputVertices BorrowStr(x,y,
  c)
proof
  let x,y,c be non pair set;
  assume
A1: not thesis;
A2: c in the carrier of BorrowStr(x,y,c) by Th6;
A3: InnerVertices BorrowStr(x,y,c) is Relation by Th1;
  x in the carrier of BorrowStr(x,y,c) & y in the carrier of BorrowStr(x,y
  ,c) by Th6;
  then
  x in InnerVertices BorrowStr(x,y,c) or y in InnerVertices BorrowStr(x,y,
  c) or c in InnerVertices BorrowStr(x,y,c) by A2,A1,XBOOLE_0:def 5;
  then
  (ex a1,b1 being object st x = [a1,b1]) or
(ex a1,b1 being object st y = [a1,b1
  ]) or ex a1,b1 being object st c = [a1,b1] by A3,RELAT_1:def 1;
  hence contradiction;
end;
