reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;
reserve D for non vertical non horizontal non empty compact Subset of TOP-REAL
  2;

theorem Th76:
  for G being TopSpace,A,B,C being Subset of G st
  A is a_component & B is a_component & C is connected & A meets C & B
  meets C holds A=B
proof
  let G be TopSpace,A,B,C be Subset of G;
  assume that
A1: A is a_component and
A2: B is a_component and
A3: C is connected and
A4: A meets C and
A5: B meets C;
A6: C /\ A={}G or C c= A by A1,A3,A4,CONNSP_1:36;
A7: C misses B or C c= B by A2,A3,CONNSP_1:36;
  per cases by A1,A2,CONNSP_1:1,34;
  suppose
    A=B;
    hence thesis;
  end;
  suppose
    A misses B;
    then
A8: A /\ B = {};
    C c= A /\ B by A4,A5,A6,A7,XBOOLE_1:19;
    then C ={} by A8;
    then C /\ A = {};
    hence thesis by A4;
  end;
end;
