reserve n for Nat,
  k for Integer;
reserve p for polyhedron,
  k for Integer,
  n for Nat;

theorem Th75:
  dim(p)-boundary(p) is one-to-one
proof
  set T = dim(p)-boundary(p);
  set U = (dim(p) - 1)-chain-space(p);
  set V = dim(p)-chain-space(p);
  set B = {p};
  assume not T is one-to-one;
  then consider x1,x2 being object such that
A1: x1 in dom T and
A2: x2 in dom T and
A3: T.x1 = T.x2 and
A4: x1 <> x2 by FUNCT_1:def 4;
  reconsider x2 as Element of V by A2;
  reconsider x1 as Element of V by A1;
  per cases by A4,Th67;
  suppose
    x1 = 0.V;
    then x2 = B & T.x1 = 0.U by A4,Th66,RANKNULL:9;
    hence thesis by A3,Th74;
  end;
  suppose
    x2 = 0.V;
    then x1 = B & T.x2 = 0.U by A4,Th66,RANKNULL:9;
    hence thesis by A3,Th74;
  end;
end;
