reserve D for non empty set;
reserve m,n,N for Nat;
reserve size for non zero Nat;
reserve f1,f2,f3,f4,f5,f6 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7 for PartialPredicate of D;
reserve d,v for object;
reserve V,A for set;
reserve z for Element of V;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve T for TypeSCNominativeData of V,A;

theorem Th1:
  loc|Seg N is one-to-one & Seg N c= dom loc implies
  for i,j being Nat st 1 <= i <= N & 1 <= j <= N & i <> j holds
  loc/.i <> loc/.j
  proof
    set f = loc|Seg N;
    assume that
A1: f is one-to-one and
A2: Seg N c= dom loc;
    let i,j be Nat such that
A3: 1 <= i <= N and
A4: 1 <= j <= N and
A5: i <> j;
A6: i in Seg N by A3,FINSEQ_1:1;
    then
A7: i in dom f by A2,RELAT_1:57;
A8: j in Seg N by A4,FINSEQ_1:1;
    then
A9: j in dom f by A2,RELAT_1:57;
A10: loc/.i = loc.i by A2,A6,PARTFUN1:def 6;
A11: f.i = loc.i by A7,FUNCT_1:47;
    f.j = loc.j by A9,FUNCT_1:47;
    hence loc/.i <> loc/.j
    by A1,A5,A7,A9,A10,A2,A8,A11,FUNCT_1:def 4,PARTFUN1:def 6;
  end;
