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 Th12:
  x = y implies reproj(i,y)*proj(1,1)=reproj(i,x)
proof
  reconsider k = proj(1,1) as Function of REAL 1,REAL;
A1: the carrier of REAL-NS n = REAL n by REAL_NS1:def 4;
  assume
A2: x=y;
A3: now
    let s be Element of REAL 1;
    reconsider r=s as Point of REAL-NS 1 by REAL_NS1:def 4;
A4: (reproj(i,y)*k).s = reproj(i,y).(k.s) by FUNCT_2:15;
    ex q be Element of REAL, z be Element of REAL n st r=<*q *> & z=x &
    reproj(i,x).r=reproj(i,z).q by Def6;
    hence reproj(i,x).s=(reproj(i,y)*k).s by A2,A4,Lm1;
  end;
  the carrier of REAL-NS 1 = REAL 1 by REAL_NS1:def 4;
  hence thesis by A1,A3,FUNCT_2:63;
end;
