reserve m, n for non zero Element of NAT;
reserve i, j, k for Element of NAT;
reserve Z for Subset of REAL 2;
reserve c for Real;
reserve I for non empty FinSequence of NAT;
reserve d1, d2 for Element of REAL;

theorem LM10:
  for X, T be Subset of REAL, f be PartFunc of REAL,REAL,
  g be PartFunc of REAL,REAL
  st X c= dom f & T c= dom g
  ex u be PartFunc of REAL 2,REAL
  st dom u = {<*x, t*> where x, t is Real: x in X & t in T}
  & for x, t be Real st x in X & t in T
  holds u/.<*x, t*> = f/.x*g/.t
  proof
    let X, T be Subset of REAL, f be PartFunc of REAL,REAL,
    g be PartFunc of REAL,REAL;
    assume X c= dom f & T c= dom g;
    defpred P1[object, object] means
    ex x, t be Real st x in X & t in T & $1 = <*x, t*> & $2 = f/.x*g/.t;
    P1: for z, w being object st z in REAL 2 & P1[z,w] holds w in REAL;
    P2: for z, w1, w2 being object st z in REAL 2 & P1[z,w1] & P1[z,w2]
    holds w1 = w2
    proof
      let z, w1, w2 be object;
      assume that
      z in REAL 2 and
      P21: P1[z,w1] and
      P22: P1[z,w2];
      consider x1,t1 be Real such that
      P23: z=<*x1,t1*> & w1=f/.x1*g/.t1 by P21;
      consider x2,t2 be Real such that
      P24: z=<*x2,t2*> & w2=f/.x2*g/.t2 by P22;
      x1 = x2 & t1 = t2 by P23, P24, FINSEQ_1:77;
      hence w1=w2 by P23, P24;
    end;
    consider u being PartFunc of REAL 2,REAL such that
    P3: for z being object holds
    (z in dom u iff (z in REAL 2 & ex w being object st P1[z,w])) and
    P4: for z being object st z in dom u holds P1[z, u.z]
    from PARTFUN1:sch 2(P1,P2);
    take u;
    for z being object holds
    (z in dom u iff z in {<*x, t*> where x, t is Real: x in X & t in T})
    proof
      let z be object;
      hereby
        assume z in dom u;
        then z in REAL 2 & ex w being object st P1[z,w] by P3;
        hence z in {<*x, t*> where x, t is Real: x in X & t in T};
      end;
      assume z in {<*x, t*> where x, t is Real: x in X & t in T};
      then consider x, t be Real such that
      P51: z = <*x, t*> & x in X & t in T;
      f/.x*g/.t in REAL;
      hence z in dom u by P3,P51,LM02;
    end;
    hence
    P5: dom u = {<*x, t*> where x, t is Real: x in X & t in T} by TARSKI:2;
    let x, t be Real;
    assume x in X & t in T; then
    P7: <*x, t*> in dom u by P5;
    then consider x1, t1 be Real such that
    x1 in X & t1 in T and
    P9: <*x, t*> = <*x1,t1*> and
    P10: u.<*x, t*> = f/.x1*g/.t1 by P4;
    x = x1 & t = t1 by P9, FINSEQ_1:77;
    hence u/. <*x, t*> = f/.x*g/.t by P10, P7, PARTFUN1:def 6;
  end;
