reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th75:
  for X being set, A being finite Subset of X, R be Order of X st
R linearly_orders A for i,j being Element of NAT st i in dom(SgmX(R,A)) & j in
  dom(SgmX(R,A)) holds SgmX(R,A)/.i = SgmX(R,A)/.j implies i = j
proof
  let X be set, A be finite Subset of X, R be Order of X;
  assume
A1: R linearly_orders A;
  let i,j be Element of NAT such that
A2: i in dom(SgmX(R,A)) and
A3: j in dom(SgmX(R,A)) and
A4: SgmX(R,A)/.i = SgmX(R,A)/.j;
  assume i <> j;
  then i < j or j < i by XXREAL_0:1;
  hence thesis by A1,A2,A3,A4,Def2;
end;
