reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th92:
  for a,b,a1,a2,a3,b1,b2,b3 being Real st
  a > 0 & b > 0 & a1 >= 1 & a2 > 0 & a3 >= 0 & b1 > 0 & b2 >= 1 & b3 >= 0 holds
  recSeqCart(a,b,a1,a2,a3,b1,b2,b3) is one-to-one
  proof
    let a,b,a1,a2,a3,b1,b2,b3 be Real such that
A1: a > 0 & b > 0 & a1 >= 1 & a2 > 0 & a3 >= 0 & b1 > 0 & b2 >= 1 & b3 >= 0;
    set f = recSeqCart(a,b,a1,a2,a3,b1,b2,b3);
    let x1,x2 be object such that
A2: x1 in dom f & x2 in dom f and
A3: f.x1 = f.x2;
    reconsider x1,x2 as Element of NAT by A2,PARTFUN1:def 2;
    now
      assume x1 <> x2;
      then x1 < x2 or x2 < x1 by XXREAL_0:1;
      then (f.x1)`1 < (f.x2)`1 or (f.x2)`1 < (f.x1)`1 by A1,Th91;
      hence contradiction by A3;
    end;
    hence thesis;
  end;
