
theorem Th5:
  for S being non empty non void ManySortedSign for A1,A2,B1,B2
  being MSAlgebra over S st the MSAlgebra of A1 = the MSAlgebra of B1 & the
  MSAlgebra of A2 = the MSAlgebra of B2 for f being ManySortedFunction of A1,A2
for g being ManySortedFunction of B1,B2 st f = g for o being OperSymbol of S st
  Args(o,A1) <> {} & Args(o,A2) <> {} for x being Element of Args(o,A1) for y
  being Element of Args(o,B1) st x = y holds f#x = g#y
proof
  let S be non empty non void ManySortedSign;
  let A1,A2,B1,B2 be MSAlgebra over S such that
A1: the MSAlgebra of A1 = the MSAlgebra of B1 & the MSAlgebra of A2 =
  the MSAlgebra of B2;
  let f be ManySortedFunction of A1,A2;
  let g be ManySortedFunction of B1,B2 such that
A2: f = g;
  let o be OperSymbol of S such that
A3: Args(o,A1) <> {} & Args(o,A2) <> {};
  let x be Element of Args(o,A1);
  let y be Element of Args(o,B1);
  assume
A4: x = y;
  f#x = (Frege(f*the_arity_of o)).x by A3,MSUALG_3:def 5;
  hence thesis by A1,A2,A3,A4,MSUALG_3:def 5;
end;
