reserve f for PartFunc of REAL-NS 1,REAL-NS 1;
reserve g for PartFunc of REAL,REAL;
reserve x for Point of REAL-NS 1;
reserve y for Real;
reserve m,n for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS n,REAL-NS 1;
reserve g for PartFunc of REAL n,REAL;
reserve x for Point of REAL-NS n;
reserve y for Element of REAL n;

theorem Th11:
  Proj(i,n)=proj(1,1)qua Function"*proj(i,n)
proof
  reconsider h = proj(1,1)qua Function" as Function of REAL,REAL 1 by Th2;
A1: the carrier of REAL-NS n = REAL n by REAL_NS1:def 4;
A2: now
    let x be Element of REAL n;
    reconsider z=x as Point of REAL-NS n by REAL_NS1:def 4;
A3: (h*proj(i,n)).x =h.(proj(i,n).x) by FUNCT_2:15;
    hence (h*proj(i,n)).x =<*proj(i,n).x*> by Lm1;
    Proj(i,n).x =Proj(i,n).z;
    then Proj(i,n).x =<*proj(i,n).x*> by Def4;
    hence Proj(i,n).x = (h*proj(i,n)).x by A3,Lm1;
  end;
  the carrier of REAL-NS 1 = REAL 1 by REAL_NS1:def 4;
  hence thesis by A1,A2,FUNCT_2:63;
end;
